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TTU Math2450 Calculus3 Sec 12.5

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    MAGDALENA TODA:
    According to my watch,
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    we are right on time to start.
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    I may be one minute
    early, or something.
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    Do you have questions out of the
    material we covered last time?
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    What I'm planning on
    doing-- let me tell you
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    what I'm planning to do.
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    I will cover triple
    integrals today.
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    And this way, you
    would have accumulated
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    enough to deal with most of
    the problems in homework four.
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    You have mastered the
    double integration by now,
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    in all sorts of coordinates,
    which is a good thing.
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    Triple integrals
    are your friend.
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    If you have understood
    the double integration,
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    you will have no
    problem understanding
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    the triple integrals.
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    The idea is the same.
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    You look at different
    domains, and then you
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    realize that there are
    Fubini-Tunelli type of results.
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    I'm going to present
    one right now.
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    And there are also regions
    of a certain type, that
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    can be treated
    differentially, and then you
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    have cases in which reversing
    the order of integration
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    for those triple integrals
    is going to help you a lot.
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    OK, 12.5 is the name of the
    section, triple integrals.
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    So what should you imagine?
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    You should imagine
    that somebody gives you
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    a function of three variables.
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    Let's call that-- it doesn't
    often have a name as a letter,
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    but let's call it w.
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    Being a function of three
    coordinates, x, y, and z,
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    where x, y, z is in R3.
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    And we have some assumptions
    about the domain D
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    that you are working
    over, and you
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    have D as a closed-bounded
    domain in R3.
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    Examples that you're going
    to do use frequently.
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    Frequently used.
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    A sphere, a ball,
    actually, because in here,
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    if a sphere is together with
    a shell, it is the ball.
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    Then you have some types of
    polyhedra in r3 of all types.
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    And by that, I mean
    the classical polyhedra
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    whose sides are just polygons.
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    But you will also have some
    curvilinear polyhedra, as well.
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    What do I mean?
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    I mean, we've seen that already.
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    For example, somebody
    give you a graph
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    of a function, g of x and y, a
    continuous function, and says,
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    OK, can you estimate the
    volume under the graph?
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    Right?
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    And until now, we treated
    this volume under the graph
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    as double integral of g of x, y,
    continuous function over d, a,
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    where d, a was dx dy.
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    And we said double integral
    over the projected domain
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    in the plane.
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    That's what I have.
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    But can I treat it, this
    volume, can I treat it
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    as a triple integral?
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    This is the question,
    and answer is--
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    so can I make three snakes?
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    The answer is yes.
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    And the way I'm
    going to define that
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    would be a triple
    integral over a 3D domain.
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    Let's call it curvilinear d in
    r3, which is the volume-- which
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    is the body under the graph
    of this positive function,
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    and above the projected
    area in plane.
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    So it's going to be a
    cylinder in this case.
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    And I'll put here 1 dv.
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    And dv is a mysterious element.
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    That's the volume element.
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    And I will talk a little
    bit about it right now.
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    So what can you imagine?
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    They give you a way to
    look at it in the book.
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    I mean, we give you a way
    to look at it in the book.
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    It's not very thorough
    in explanations,
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    but it certainly gives you the
    general idea of what you want,
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    what you need.
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    So somebody gives you a potato.
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    It doesn't have to
    be this cylinder.
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    It's something beautiful, a
    body inside a compact surface.
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    Let's say there are
    no self-intersections.
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    You have some compact surface,
    like a sphere or a polyhedron,
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    assume it's simply
    connected, and it doesn't
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    have any self-intersections.
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    So a beautiful
    potato that's smooth.
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    If you imagine a potato
    that has singularities,
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    like most potatoes have
    singularities, boo-boos,
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    and cuts, so that's bad.
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    So think about some
    regular surface
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    that's closed, no
    self-intersections,
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    and that is a potato
    that is [INAUDIBLE].
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    Oh let's call it--
    p for potato, no,
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    because I have got to
    use p for the partition.
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    So let me call it
    D from 3D domain,
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    because it's a three-dimensional
    domain, enclosed
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    by a curve, enclosed
    by a compact surface.
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    So think potato.
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    What do we do in
    terms of partitions?
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    Those pixels were pixels for
    the 2D world in flat line.
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    But now, we don't
    have pixels anymore.
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    Yes we do.
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    I was watching lots of
    sci-fi, and the holograms
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    have the
    three-dimensional pixels.
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    I'm going to try and
    make a partition.
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    It's going to be a hard way
    to partition this potato.
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    But you have to imagine you
    have a rectangular partition,
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    so every little pixel
    will be a-- is not cube.
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    It has to be a little
    tiny parallelepiped.
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    So a 3D pixel, let me
    put pixel in quotes,
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    because this is
    kind of the idea.
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    Well have what
    kind of dimensions?
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    We'll have three
    dimensions, right?
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    Three dimensions, a delta xk,
    a delta yk, and a delta zk
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    for the pixel number k.
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    That's pixel number k.
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    We have to number them,
    see how many they are.
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    Where k is from 1 to n, and
    is the total number of 3D
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    pixels that I'm covering
    the whole thing.
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    So don't think graphing
    paper, anymore,
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    because that's outdated.
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    Don't even think of 2D image.
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    Think of some hologram,
    where you cover everything
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    with tiny, tiny,
    tiny 3D pixels so
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    that in the limit, when you pass
    to the limit, with respect to n
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    and going to infinity,
    the discrete image,
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    you're going to have something
    like a diamond shaped thingy,
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    will convert to
    the smooth potato.
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    So as n goes to
    infinity, that surface
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    made of tiny, tiny squares
    will convert to the data.
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    So how do you actually find
    a triple snake integral f
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    of x, y, z over the
    domain D, dx dy dz.
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    OK, this theoretically
    should be what?
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    Think pixels.
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    Limit as n goes to infinity
    of the sum of the--
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    what do I need to do?
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    Think the whole
    partition into pixels.
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    How many pixels? n pixels
    total is called a p.
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    Script p.
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    And the normal p will be the
    highest diameter of-- highest
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    diameter among all pixels.
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    And you're going to
    say, Oh, what the heck?
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    I don't understand it.
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    I have these three-dimensional
    cubes, or three-dimensional
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    barely by p that get tinier,
    and tinier, and tinier.
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    What in the world is going to
    be a diameter of such a pixel?
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    Well, you have to
    take this pixel
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    and magnify it so we can look
    at it a little bit better.
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    What do we mean by
    diameter of this pixel?
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    Let's call this pixel k.
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    The maximum of all the distances
    you can compute between two
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    arbitrary points inside.
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    Between two arbitrary
    points inside,
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    you have many [INAUDIBLE].
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    So the maximum of the
    distance between points,
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    let's call them r and
    q inside the pixel.
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    So if it were for me
    to ask you to find
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    that diameter in this
    case, what would that be?
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    STUDENT: It's the diagonal.
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    MAGDALENA TODA: It's the
    diagonal between this corner
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    and the opposite corner.
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    So this would be the highest
    distance inside this pixel.
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    If it's a cube-- you see
    that I wanted a cube.
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    If it's a parallelepiped,
    it's the same idea.
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    So I have that opposite
    corner distance kind of thing.
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    OK.
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    So I know what I want.
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    I want n to go to infinity.
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    That means I'm going to
    have the p going to 0.
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    The length of the
    highest diameter
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    will go shrinking to 0.
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    And then I'm going to say
    here, what do I have inside?
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    F of some intermediate point.
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    In every pixel, I take a point.
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    Another pixel, another
    point, and so on.
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    So how many such
    points do I have?
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    n, because it's the
    number of pixels.
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    So inside, let's call
    this as pixel p k.
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    What is the little point that
    I took out of the [INAUDIBLE]
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    inside the cube, or
    inside the pixel?
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    Let's call that mister x k
    star, y k star, x k star.
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    Why do we put a star?
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    Because he is a star.
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    He wants to be number one
    in these little domains,
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    and says I'm a star.
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    So we take that intermediate
    point, x k star,
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    y k star, x k star.
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    Then we have this function
    multiplied by the delta v k.
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    somebody tell me what
    this delta v k will mean,
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    because ir really looks weird.
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    And then k will be from 1 to n.
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    So what do you do?
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    You sum up all these
    weighted volumes.
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    This is a weight.
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    So this is a volume.
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    All these weighted volumes.
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    We sum them up for all
    the pixels k from 1 to n.
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    We are going to get
    something like this cover
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    with tiny parallelepipeds
    in the limit,
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    as the partitions' [? norm ?]
    go to 0, or the number of pixels
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    goes to infinity.
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    This discrete
    surface will converge
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    to the beautiful smooth
    potato, and give you
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    a perfect linear image.
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    Actually, if we saw a
    hologram, this is what it is.
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    Our eyes actually
    see a bunch of I
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    tiny-- many, many, many,
    many, millions of pixels
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    that are cubes in 3D.
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    But it's an optical illusion.
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    We see, OK, it's a curvilinear,
    it's a smooth body of a person.
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    It's not smooth at all.
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    If you get closer and
    closer to that diagram
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    and put your eye
    glasses on, you are
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    going to see, oh, this
    is not a real person.
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    It's made of pixels
    that are all cubes.
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    just the same, you see your
    digital image of your picture
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    on Facebook, whatever it is.
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    If you would be able
    to be enlarge it,
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    you would see the pixels, being
    little tiny squares there.
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    The graphical
    imaging has improved.
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    The quality of our digital
    imaging has improved a lot.
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    But of course, 20 years ago,
    when you weren't even born,
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    we could still see the pixels
    in the photographic images
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    in a digital camera.
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    Those tiny first cameras,
    what was that, '98?
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    STUDENT: Kodak.
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    MAGDALENA TODA: Like AOL
    cameras that were so cheap.
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    The cheaper the camera,
    the worse the resolution.
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    I remember some resolutions
    like 400 by 600.
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    STUDENT: Black and white.
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    MAGDALENA TODA: Not
    black and white.
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    Black and white
    would have been neat.
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    But really nasty in the sense
    that you had the feeling
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    that the colors
    were not even-- they
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    were blending into each
    other, because the resolution
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    was so small.
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    So it was not at all
    pleasing to the eye.
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    What was good is that
    any kind of defects you
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    would have, something
    like a pimple
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    could not be seen in that,
    because the resolution was
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    so slow that you couldn't
    see the boo-boos,
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    the pimples, the defects
    of a face or something.
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    Now, you can see.
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    With the digital cameras
    we have now, we can do,
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    of course, Adobe
    Photoshop, and all of us
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    will look great if we
    photoshop our pictures.
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    OK.
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    So this is what it
    is in the limit.
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    But in reality, in
    the everyday reality,
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    you cannot take Riemann
    sums like that--
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    this is a Riemann
    approximating sum--
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    and then cast to the limit, and
    get ideal curvilinear domains.
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    No.
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    You don't do that.
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    You have to deal
    with the equivalent
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    of the fundamental
    theorem of calculus
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    from Calc I, which is called the
    Fubini-Tonelli type of theorem
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    in Calc III.
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    So say it again.
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    So the Fubini-Tunelli
    theorem that you
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    learned for double
    integrals over a rectangle
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    can be generalized to the
    Fubini-Tonelli theorem
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    over a parallelepiped.
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    And it's the same
    thing, practically,
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    as applying the fundamental
    theorem of calculus in Calc I.
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    So somebody says, well, let me
    start with a simple example.
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    I give you a-- you
    will say, Magdalena,
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    you are offending us.
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    This is way too easy.
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    What do you think, that we
    cannot understand the concept?
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    I'll just try to start with
    the simplest possible example
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    that I think of.
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    So x is between a and b.
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    In my case, they will
    be positive numbers,
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    because I want everything
    to be in the first octant.
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    First octant means x positive,
    y positive, and z positive
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    all together.
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    To make my life easier,
    I take that example,
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    and I say, I know the
    numbers for x, y, z.
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    I would like you to
    compute two integrals.
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    One would be the
    volume of this object.
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    Let's call it body.
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    It's not a dead body,
    it's just body in 3D.
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    The volume of the
    body, we say it
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    like a mathematician, V of B.
    What is that by definition?
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    Who's going to tell me?
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    Triple snake.
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    Don't say triple
    snake to other people,
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    because other professors
    are more orthodox than me.
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    They will laugh-- they
    will not joke about it.
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    So triple integral over the
    body of-- to get the volume,
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    the weight must be 1.
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    f integral must be 1, and
    then you have exactly dV.
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    How can I convince
    you what we have here,
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    in terms of Fubini-Tonelli?
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    It's really beautiful.
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    B is going to be a, b segment
    cross product, c, d segment,
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    cross product.
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    What is the altitude?
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    E, f, e to f.
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    Interval e to f
    means the height.
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    So length, width, height.
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    This is the box, or carry-on,
    or USPS parcel, or whatever
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    box you want to measure.
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    So how am I going to set up
    the Fubini-Tonelli integral?
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    a to b, c to d, e to f, 1.
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    And now, who counts first?
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    dz, dy, dx.
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    So it is like the equivalent
    of the vertical strip
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    thingy in double corners,
    double integrals.
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    Yes, sir?
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    STUDENT: Professor,
    why did you use 1 dV?
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    Why 1?
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    MAGDALENA TODA: OK.
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    You'll see in a second.
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    This is the same
    thing we do for areas.
  • 20:09 - 20:14
    So when you compute an
    area-- very good question.
  • 20:14 - 20:18
    If you use 1 here, and
    you put delta [? ak ?]
  • 20:18 - 20:21
    that is the graphing paper area.
  • 20:21 - 20:25
    It's going to be
    all the tiny areas,
  • 20:25 - 20:28
    summed up, sum of all
    the delta [? ak ?] which
  • 20:28 - 20:31
    means this little pixel,
    plus this little pixel,
  • 20:31 - 20:33
    plus this little pixel,
    plus this little pixel,
  • 20:33 - 20:38
    plus 1,000 pixels all together
    will cover up the area.
  • 20:38 - 20:42
    If you have the
    volume of a potato,
  • 20:42 - 20:44
    a body that is alive,
    but shouldn't move.
  • 20:44 - 20:47
    OK, it should stay in one place.
  • 20:47 - 20:50
    Then, to compute the
    volume of the potato,
  • 20:50 - 20:54
    you have to say, the
    potato, the smooth potato,
  • 20:54 - 20:59
    is the limit of the sum of
    all the tiny cubes of potato,
  • 20:59 - 21:03
    if you cut the potato in many
    cubes, like you cut cheese.
  • 21:03 - 21:06
    They got a bunch of cheddar
    cheese into small cubes,
  • 21:06 - 21:11
    and they feed us with crackers
    and wine-- OK, no comments.
  • 21:11 - 21:15
    So you have delta vk, you
    have 1,000 little cubes,
  • 21:15 - 21:18
    tiny, tiny, tiny,
    like that Lego.
  • 21:18 - 21:20
    OK, forget about the cheese.
  • 21:20 - 21:23
    The cheese cubes
    are way too big.
  • 21:23 - 21:26
    So imagine Legos that
    are really performing
  • 21:26 - 21:29
    with millions of little pieces.
  • 21:29 - 21:36
    Have you seen the exhibit, Lego
    exhibit with almost invisible
  • 21:36 - 21:40
    Legos at the Civic Center?
  • 21:40 - 21:43
    They have that art festival.
  • 21:43 - 21:46
    How many of you go
    to the art festival?
  • 21:46 - 21:49
    Is it every April?
  • 21:49 - 21:51
    Something like that.
  • 21:51 - 21:54
    So imagine those little
    tiny Legos, but being cubes
  • 21:54 - 21:55
    and put together.
  • 21:55 - 21:56
    This is what it is.
  • 21:56 - 21:58
    So f Vy.
  • 21:58 - 22:02
    Now, can we verify
    the volume of a box?
  • 22:02 - 22:02
    It's very easy.
  • 22:02 - 22:04
    What do we do?
  • 22:04 - 22:08
    Well, first of all, I
    would do it in a slow way,
  • 22:08 - 22:10
    and you are going to
    shout at me, I know.
  • 22:10 - 22:14
    But I'll tell you why
    you need to bear with me.
  • 22:14 - 22:17
    So integral of 1 dz goes first.
  • 22:17 - 22:19
    That's z between f and e.
  • 22:19 - 22:21
    So it's f minus e, am I right?
  • 22:21 - 22:23
    You say, duh, that's
    to easy for me.
  • 22:23 - 22:26
    I'm know it's too
    easy for me, but I'm
  • 22:26 - 22:27
    going somewhere with it.
  • 22:27 - 22:30
    dy dx.
  • 22:30 - 22:35
    The one inside, f minus e
    is a constant, pulls out,
  • 22:35 - 22:37
    completely out of the product.
  • 22:37 - 22:43
    And then I have integral from
    a to b of-- what is that left?
  • 22:43 - 22:47
    1 dy, y between d and c.
  • 22:47 - 22:48
    So d minus c, right?
  • 22:48 - 22:53
    d minus c dx.
  • 22:53 - 22:57
    And so on and so
    forth, until I get it.
  • 22:57 - 23:02
    If minus c times d
    minus c times b minus a,
  • 23:02 - 23:06
    and goodbye, because this
    is the volume of the box.
  • 23:06 - 23:08
    It's the height.
  • 23:08 - 23:10
    This is the height.
  • 23:10 - 23:13
    No, excuse me, guys.
  • 23:13 - 23:16
    The height is-- this
    one is the height.
  • 23:16 - 23:21
    This is the width, and this is
    the length, whatever you want.
  • 23:21 - 23:22
    All right.
  • 23:22 - 23:25
    How could I have done it if
    I were a little bit smarter?
  • 23:25 - 23:28
    STUDENT: You could have just
    put it in three integrals.
  • 23:28 - 23:30
    MAGDALENA TODA:
    Right Hey, I have
  • 23:30 - 23:36
    a theorem, just like
    before, which says
  • 23:36 - 23:38
    three integrals in a product.
  • 23:38 - 23:40
    This is what Matt
    immediately remembered.
  • 23:40 - 23:44
    We had two integrals
    in a product last time.
  • 23:44 - 23:47
    So what have we proved
    in double integrals
  • 23:47 - 23:50
    remains valid in
    triple integrals
  • 23:50 - 23:53
    if we have something like that.
  • 23:53 - 23:55
    So I'm going the same theorem.
  • 23:55 - 23:56
    It's in the book.
  • 23:56 - 23:57
    We have a proof.
  • 23:57 - 24:02
    So you have integral from
    a to b, c to d, e to f.
  • 24:02 - 24:08
    And then, some guys that you
    like, f of x, times g of y,
  • 24:08 - 24:11
    times h of z.
  • 24:11 - 24:16
    Functions of x, y, z,
    separated variables.
  • 24:16 - 24:19
    So f, a function of
    x only, g a function
  • 24:19 - 24:21
    of y only, h a
    function of z only.
  • 24:21 - 24:23
    This is the complicated case.
  • 24:23 - 24:26
    And then I have
  • 24:26 - 24:28
    STUDENT: dz, dy, dx.
  • 24:28 - 24:29
    MAGDALENA TODA: dz, dy, dx.
  • 24:29 - 24:30
    Excellent.
  • 24:30 - 24:33
    Thanks for whispering,
    because I was a little bit
  • 24:33 - 24:36
    confused for a second.
  • 24:36 - 24:39
    So, just as Matt said,
    go ahead and observe
  • 24:39 - 24:42
    that you can treat them one
    at a time like you did here,
  • 24:42 - 24:47
    and integrate one at a time, and
    integrate again, and pull out
  • 24:47 - 24:50
    a constant, integrate
    again, pull out a constant.
  • 24:50 - 24:53
    But practically this is
    exactly the same as integral
  • 24:53 - 25:02
    from a to b of f of x
    alone, dx, times integral
  • 25:02 - 25:12
    from c to d, g of y alone,
    dy, and times integral from e
  • 25:12 - 25:18
    to f of h of z, dz, and close.
  • 25:18 - 25:22
    So you've seen the version
    of the double integral,
  • 25:22 - 25:27
    and this is the same result
    for triple integrals.
  • 25:27 - 25:30
    And it's practically--
    what is the proof?
  • 25:30 - 25:34
    You just pull out one
    at a time, so the proof
  • 25:34 - 25:38
    is that you start working and
    say, mister z counts here,
  • 25:38 - 25:40
    and he's the only
    one that counts.
  • 25:40 - 25:44
    These guys get out for a
    walk one at a time outside
  • 25:44 - 25:46
    of the first integral inside.
  • 25:46 - 25:49
    And then, integral
    of h of z, dz,
  • 25:49 - 25:53
    over the corresponding domain,
    will be just a constant, c1,
  • 25:53 - 25:55
    that pulls out.
  • 25:55 - 26:00
    And that is that--
    c1 that pulls out.
  • 26:00 - 26:03
    Ans since you pull them out
    in this product one at a time,
  • 26:03 - 26:05
    that's what you get.
  • 26:05 - 26:09
    I'm not going to give you this
    as an exercise in the midterm
  • 26:09 - 26:12
    with a proof, but this is
    one of the first exercises
  • 26:12 - 26:16
    I had as a freshman
    in my multi--
  • 26:16 - 26:20
    I took it as a freshman,
    as multivariable calculus.
  • 26:20 - 26:23
    And it was a pop quiz.
  • 26:23 - 26:27
    My professor just came
    one day, and said, guys,
  • 26:27 - 26:30
    you have to try to do this
    [? before ?] by yourself.
  • 26:30 - 26:33
    And some of us did,
    some of us didn't.
  • 26:33 - 26:35
    To me, it really
    looked very easy.
  • 26:35 - 26:41
    I was very happy to prove it,
    in an elementary way, of course.
  • 26:41 - 26:41
    OK.
  • 26:41 - 26:46
    So how hard is it
    to generalize, to go
  • 26:46 - 26:48
    to non-rectangular domains?
  • 26:48 - 26:49
    Of course it's a pain.
  • 26:49 - 26:54
    It's really a pain,
    like it was before.
  • 26:54 - 27:00
    But you will be able to
    figure out what's going on.
  • 27:00 - 27:03
    In most cases,
    you're going to have
  • 27:03 - 27:07
    a domain that's really
    not bad, a domain that
  • 27:07 - 27:09
    has x between fixed values.
  • 27:09 - 27:14
    For example y between
    your favorite guys,
  • 27:14 - 27:20
    something like f of x and
    g of x, top and bottom.
  • 27:20 - 27:22
    That's what you had
    for double integral.
  • 27:22 - 27:26
    Well, in addition,
    in this case, you
  • 27:26 - 27:31
    will have z between--
    let's make this guy
  • 27:31 - 27:36
    big F and big G,
    other functions.
  • 27:36 - 27:38
    This is going to be
    a function of x, y.
  • 27:38 - 27:41
    This is going to be
    a function of x, y,
  • 27:41 - 27:44
    and that's the
    upper and the lower.
  • 27:44 - 27:50
    And find the triple integral
    of, let's say 1 over d dV
  • 27:50 - 27:53
    will be a volume of the potato.
  • 27:53 - 27:56
    Now, I'm sick of potatoes,
    because they're not
  • 27:56 - 27:59
    my favorite food.
  • 27:59 - 28:03
    Let me imagine I'm making a
    tetrahedron, a lot of cheese.
  • 28:03 - 28:09
  • 28:09 - 28:14
    I'm going to draw this same
    tetrahedron from last time.
  • 28:14 - 28:15
    So what did we do last time?
  • 28:15 - 28:18
    We took a plane
    that was beautiful,
  • 28:18 - 28:22
    and we said let's
    cut with that plane.
  • 28:22 - 28:25
    This is the plane we are
    cutting the cheese with.
  • 28:25 - 28:26
    It's a knife.
  • 28:26 - 28:28
    x plus y plus z equals 1.
  • 28:28 - 28:30
    Imagine that there's
    an infinite knife that
  • 28:30 - 28:32
    comes into the frame.
  • 28:32 - 28:33
    Everything is cheese.
  • 28:33 - 28:37
    The space, the universe is
    covered in solid cheese.
  • 28:37 - 28:42
    So the whole thing,
    the Euclidean space
  • 28:42 - 28:44
    is covered in cheddar cheese.
  • 28:44 - 28:45
    That's all there.
  • 28:45 - 28:48
    From everywhere, you
    come with this knife,
  • 28:48 - 28:53
    and you cut along
    this plane-- hi
  • 28:53 - 28:55
    let's call this
    [? high plane. ?]
  • 28:55 - 29:00
    And then you cut the x
    plane along the x, y plane,
  • 29:00 - 29:03
    y, z plane and x, z plane.
  • 29:03 - 29:04
    What are these called?
  • 29:04 - 29:07
    Planes of coordinates.
  • 29:07 - 29:07
    And what do you obtain?
  • 29:07 - 29:09
    Then, you throw
    everything away, and you
  • 29:09 - 29:15
    maintain only the
    tetrahedron made of cheese.
  • 29:15 - 29:18
    Now, you remember
    what the corners were.
  • 29:18 - 29:19
    This is 0, 0, 0.
  • 29:19 - 29:20
    It's a piece of cake.
  • 29:20 - 29:23
    But I want to know the vertices.
  • 29:23 - 29:28
    And you know them, and I
    don't want to spend time
  • 29:28 - 29:30
    discussing why you know them.
  • 29:30 - 29:31
    So
  • 29:31 - 29:31
    STUDENT: 0--
  • 29:31 - 29:32
    MAGDALENA TODA: 1, 0, 0.
  • 29:32 - 29:33
    Thank you.
  • 29:33 - 29:34
    Huh?
  • 29:34 - 29:35
    STUDENT: 0, 1, 0.
  • 29:35 - 29:36
    MAGDALENA TODA: Yes.
  • 29:36 - 29:38
    And 0, 0, 1.
  • 29:38 - 29:40
    All right.
  • 29:40 - 29:40
    Great.
  • 29:40 - 29:44
    The only thing is, if we see
    the cheese being a solid,
  • 29:44 - 29:50
    we don't see this part, the
    three axes of corners behind.
  • 29:50 - 29:56
    so I'm going to make them
    dotted, and you see the slice,
  • 29:56 - 29:59
    here, it has to
    be really planar.
  • 29:59 - 30:01
    And you ask yourself,
    how do you set up
  • 30:01 - 30:04
    the triple integral
    that represents
  • 30:04 - 30:06
    the volume of this object?
  • 30:06 - 30:07
    Is it hard?
  • 30:07 - 30:09
    It shouldn't be hard.
  • 30:09 - 30:13
    You just have to think what
    the domain will be like,
  • 30:13 - 30:17
    and you say the domain is
    inside the tetrahedron.
  • 30:17 - 30:18
    Do you want d or t?
  • 30:18 - 30:21
    T from tetrahedron.
  • 30:21 - 30:22
    It doesn't matter.
  • 30:22 - 30:23
    We have a new name.
  • 30:23 - 30:27
    We get bored of all sorts
    of names and notations.
  • 30:27 - 30:28
    We change them.
  • 30:28 - 30:34
    Mathematicians have imagination,
    so we change our notations.
  • 30:34 - 30:35
    Like we cannot change
    our identities,
  • 30:35 - 30:38
    and we suffer because of that.
  • 30:38 - 30:41
    So you can be a nerd
    mathematician imagining
  • 30:41 - 30:44
    you're Spiderman,
    and you can take,
  • 30:44 - 30:48
    give any name you want, and
    you can adopt a new name,
  • 30:48 - 30:52
    and this is behind
    our motivation
  • 30:52 - 30:56
    why we like to change names
    and change notation so much.
  • 30:56 - 30:57
    OK?
  • 30:57 - 31:03
    So we have triple integral of
    this T. All right, of what?
  • 31:03 - 31:06
    1 dV.
  • 31:06 - 31:07
    Good.
  • 31:07 - 31:10
    Now we understand
    what we need to do,
  • 31:10 - 31:12
    just like [? Miteish ?]
    asked me why.
  • 31:12 - 31:16
    OK, now we know this is going
    to be a limit of little cubes.
  • 31:16 - 31:18
    If were to cover
    this piece of cheese
  • 31:18 - 31:22
    in tiny, tiny,
    infinitesimally small cubes.
  • 31:22 - 31:25
    But now we know a
    method to do it.
  • 31:25 - 31:30
    So according to--
    Fubini-Tonelli type of result.
  • 31:30 - 31:39
    We would have a between--
    no, x-- is first, dz.
  • 31:39 - 31:43
    z is first, y is moving
    next, x is moving last.
  • 31:43 - 31:47
    z is constrained to
    move between a and b.
  • 31:47 - 31:51
    But in this case, a and b should
    be prescribed by you guys,
  • 31:51 - 31:56
    because you should think
    where everybody lives.
  • 31:56 - 32:00
    Not you, I mean the coordinates
    in their imaginary world.
  • 32:00 - 32:03
    The coordinates
    represent somebodies.
  • 32:03 - 32:04
    STUDENT: 0.
  • 32:04 - 32:08
    MAGDALENA TODA: x, 0 to 1.
  • 32:08 - 32:11
    How should I give you
    a feeling for that?
  • 32:11 - 32:12
    Just draw this line.
  • 32:12 - 32:15
    This red segment between 0 to 1.
  • 32:15 - 32:18
    That expresses everything
    instead of words
  • 32:18 - 32:23
    into pictures, because every
    picture is worth 1,000 words.
  • 32:23 - 32:27
    y is married to
    x, unfortunately.
  • 32:27 - 32:31
    y cannot say, oh, I am y,
    I'm going wherever I want.
  • 32:31 - 32:34
    He hits his head against
    this purple line.
  • 32:34 - 32:37
    He cannot go beyond
    that purple line.
  • 32:37 - 32:40
    He's constrained, poor y.
  • 32:40 - 32:41
    So he says, I'm moving.
  • 32:41 - 32:42
    I'm mister y.
  • 32:42 - 32:46
    I'm moving in this direction,
    but I cannot go past the purple
  • 32:46 - 32:49
    line in plane here.
  • 32:49 - 32:52
  • 32:52 - 32:57
    I need you, because
    if you go, I'm lost.
  • 32:57 - 32:59
    y is between 0 and--
  • 32:59 - 33:00
    STUDENT: 1 minus x.
  • 33:00 - 33:01
    MAGDALENA TODA: 1 minus x.
  • 33:01 - 33:02
    Excellent Roberto.
  • 33:02 - 33:05
    How did we think about this?
  • 33:05 - 33:08
    The purple line has
    equation-- how do you
  • 33:08 - 33:11
    get to the equation of the
    purple line, first of all?
  • 33:11 - 33:15
    In your imagination,
    your plug in z equals 0.
  • 33:15 - 33:19
    So the purple line would
    be x plus y equals 1.
  • 33:19 - 33:25
    And so mister y will
    be 1 minus x here.
  • 33:25 - 33:28
    That's how you got it.
  • 33:28 - 33:32
    And finally, z is that--
    mister z foes from the floor
  • 33:32 - 33:34
    all the way--
    imagine somebody who
  • 33:34 - 33:40
    is like-- z is a
    helium balloon, and he
  • 33:40 - 33:43
    is left-- you let him
    go from the floor,
  • 33:43 - 33:45
    and he goes all the
    way to the ceiling.
  • 33:45 - 33:49
    And the ceiling is not
    flat like our ceiling.
  • 33:49 - 33:55
    The ceiling is
    this oblique plane.
  • 33:55 - 33:59
    So z is going to hit his head
    against the roof at some point,
  • 33:59 - 34:01
    and he doesn't know
    where he is going
  • 34:01 - 34:06
    to hit his head, unless you
    tell him where that happens.
  • 34:06 - 34:11
    So he knows he leaves at 0,
    and he's going to end up where?
  • 34:11 - 34:13
    STUDENT: 1 minus y minus x.
  • 34:13 - 34:14
    MAGDALENA TODA: Excellent.
  • 34:14 - 34:16
    1 minus x minus y.
  • 34:16 - 34:18
    How do we do that?
  • 34:18 - 34:23
    We pull z out of that, and
    say, 1 minus x minus y.
  • 34:23 - 34:27
    So that is the equation of
    the shaded purple plane,
  • 34:27 - 34:30
    and this is as
    far as you can go.
  • 34:30 - 34:34
    You cannot go past the
    roof of your house,
  • 34:34 - 34:38
    which is the purple plane,
    the purple shaded plane.
  • 34:38 - 34:40
    So here you are.
  • 34:40 - 34:40
    Is this hard?
  • 34:40 - 34:41
    No.
  • 34:41 - 34:44
    In many problems on the
    final and on the midterm,
  • 34:44 - 34:48
    we tell you, don't even
    think about solving that,
  • 34:48 - 34:53
    because we believe you.
  • 34:53 - 34:56
    Just set up the integral.
  • 34:56 - 34:59
    I might give you something
    like that again, just
  • 34:59 - 35:04
    set up the integral and
    you have to do that.
  • 35:04 - 35:08
    But now, I would like
    to actually work it out,
  • 35:08 - 35:12
    see how hard it is.
  • 35:12 - 35:15
    So is this hard
    to work this out?
  • 35:15 - 35:18
  • 35:18 - 35:21
    I have to do it one at
    a time, because you see,
  • 35:21 - 35:24
    I don't have fixed endpoints.
  • 35:24 - 35:28
    I cannot say, I'm applying the
    problem with the integral if f
  • 35:28 - 35:32
    times the integral of g, times--
    so I have to integrate one
  • 35:32 - 35:38
    at a time, because I don't
    have fixed endpoints.
  • 35:38 - 35:42
    And the integral of 1dz is
    z between that and that.
  • 35:42 - 35:47
    So z, 1 minus x minus y
    will be what's left over,
  • 35:47 - 35:50
    and then I have dy,
    and then I have dx.
  • 35:50 - 35:54
    And at this point it
    looks horrible enough,
  • 35:54 - 35:56
    but we have to pray
    that in the end
  • 35:56 - 36:02
    it's not going to be so hard,
    and I'm going to keep going.
  • 36:02 - 36:05
    So we have integral from 0 to 1.
  • 36:05 - 36:12
    We have integral
    from 0 to 1 minus x..
  • 36:12 - 36:17
    I'll just copy and paste it.
  • 36:17 - 36:20
    Which is integral from 0 to 1.
  • 36:20 - 36:25
    Now I have to think,
    and that's dangerous.
  • 36:25 - 36:27
    I have 1 minux x
    with respect to y.
  • 36:27 - 36:29
    This is going to be ugly.
  • 36:29 - 36:34
    That's a constant with
    respect to y, and times y,
  • 36:34 - 36:39
    minus-- integrate with
    respect to y, y is [? what? ?]
  • 36:39 - 36:40
    y squared over 2.
  • 36:40 - 36:46
  • 36:46 - 36:49
    Between y equals 0 down.
  • 36:49 - 36:52
    That's going to save my
    life, because for y equals 0,
  • 36:52 - 36:55
    0 is going to be a
    great simplification.
  • 36:55 - 37:00
    And for y equals
    1 minus x on top,
  • 37:00 - 37:02
    hopefully it's not going
    to be the end of the world.
  • 37:02 - 37:07
    It looks ugly now, but
    I'm an optimistic person,
  • 37:07 - 37:11
    so I hope that this is
    going to get better.
  • 37:11 - 37:14
    And I can see it's
    going to get better.
  • 37:14 - 37:16
    So I have integral
    from here to 1.
  • 37:16 - 37:18
    And now I say, OK, let me think.
  • 37:18 - 37:20
    Life is not so bad.
  • 37:20 - 37:21
    Why?
  • 37:21 - 37:25
    1 minus x, 1 minus x
    is 1 minus x squared.
  • 37:25 - 37:28
    I could think faster, you
    could think faster than me,
  • 37:28 - 37:30
    but I don't want to rush.
  • 37:30 - 37:36
    1 minus x squared over 2.
  • 37:36 - 37:37
    So it's not bad at all.
  • 37:37 - 37:43
    Look, I'm getting this guy who
    is beautiful in the end, when
  • 37:43 - 37:45
    I'm going to
    integrate, and you have
  • 37:45 - 37:48
    to keep your fingers
    crossed for me,
  • 37:48 - 37:54
    because I don't know
    what I'm going to get.
  • 37:54 - 38:03
    So I get integral from 0 to 1,
    1/2 out, 1 minus x squared dx.
  • 38:03 - 38:06
    Is this bad?
  • 38:06 - 38:08
    Can you do this by
    yourself without my help?
  • 38:08 - 38:11
    What are you going to do?
  • 38:11 - 38:16
    x squared minus 2x plus 1.
  • 38:16 - 38:17
    That's the square.
  • 38:17 - 38:19
    STUDENT: Why not just change it?
  • 38:19 - 38:20
    MAGDALENA TODA: Huh?
  • 38:20 - 38:22
    STUDENT: Why not just change it?
  • 38:22 - 38:24
    MAGDALENA TODA: You
    can do it in many ways.
  • 38:24 - 38:26
    You can do whatever you want.
  • 38:26 - 38:28
    I don't care.
  • 38:28 - 38:32
    I want you to the right
    answer one way or another.
  • 38:32 - 38:35
    So I'm going to clean a
    little bit around here.
  • 38:35 - 38:40
  • 38:40 - 38:41
    It's dirty.
  • 38:41 - 38:42
    You do it.
  • 38:42 - 38:47
    You have one minute
    and a half to finish.
  • 38:47 - 38:50
    And tell me what you get.
  • 38:50 - 38:52
    STUDENT: 1 minus x
    cubed over six negative.
  • 38:52 - 38:53
    MAGDALENA TODA: No, no.
  • 38:53 - 38:55
    In the end is the number.
  • 38:55 - 38:57
    What number?
  • 38:57 - 38:59
    But you have to go slow.
  • 38:59 - 39:02
    I need three people to
    give me the same answer.
  • 39:02 - 39:04
    Because then it's like in that
    proverb, if two people tell
  • 39:04 - 39:06
    you drunk, you go to bed.
  • 39:06 - 39:10
    I need three people to tell
    me what the answer is in order
  • 39:10 - 39:13
    to believe them.
  • 39:13 - 39:14
    Three witnesses.
  • 39:14 - 39:16
    STUDENT: 1 [? by ?] 6.
  • 39:16 - 39:19
    MAGDALENA TODA: Who got
    1 over 6, raise hand?
  • 39:19 - 39:20
    Wow, guys, you're fast.
  • 39:20 - 39:23
    Can you raise hands again?
  • 39:23 - 39:26
    OK, being fast doesn't
    mean you're the best,
  • 39:26 - 39:30
    but I agree you do a very good
    job, all of you in general.
  • 39:30 - 39:35
    So I believe there were
    eight people or nine people.
  • 39:35 - 39:36
    1 over 6.
  • 39:36 - 39:41
    Now, how could I have cheated
    on this problem on the final?
  • 39:41 - 39:43
    STUDENT: It's a
    [? junction ?] from this--
  • 39:43 - 39:44
    MAGDALENA TODA: Right.
  • 39:44 - 39:48
    In this case, being a volume,
    I would have been lucky enough,
  • 39:48 - 39:51
    and say, it is the
    volume of a tetrahedron.
  • 39:51 - 39:56
    I go, the tetrahedron
    has area of the base 1/2,
  • 39:56 - 39:57
    the height is 1.
  • 39:57 - 40:01
    1/2 times 1 divided by 3 is 1/6.
  • 40:01 - 40:06
    And just pretend on the
    final that I actually
  • 40:06 - 40:07
    computed everything.
  • 40:07 - 40:12
    I could have done that, from
    here jump to here, or from here
  • 40:12 - 40:13
    jump straight to here.
  • 40:13 - 40:16
    And ask you, how did you
    get from here to here?
  • 40:16 - 40:19
    And you say, I'm a genius.
  • 40:19 - 40:20
    Could I not believe you?
  • 40:20 - 40:23
    I have to give you full credit.
  • 40:23 - 40:28
    However, what would you
    have done if I said compute,
  • 40:28 - 40:31
    I don't know, something
    worse, something
  • 40:31 - 40:37
    like triple integral of x,
    y, z over the tetrahedron 2.
  • 40:37 - 40:40
    In that case, you cannot cheat.
  • 40:40 - 40:42
    You're not lucky
    enough to cheat.
  • 40:42 - 40:44
    You're lucky enough
    to cheat when
  • 40:44 - 40:47
    you have a volume
    of a prism, you
  • 40:47 - 40:50
    have a volume of-- and volume
    means this should be the number
  • 40:50 - 40:52
    1 here, number 1.
  • 40:52 - 40:56
    So if you have number 1, here,
    or I ask you for the volume,
  • 40:56 - 40:59
    and it's a prism, or
    tetrahedron, or sphere,
  • 40:59 - 41:02
    or something, go
    ahead and cheat,
  • 41:02 - 41:04
    and pretend that you're
    actually solving the integral.
  • 41:04 - 41:05
    Yes, sir.
  • 41:05 - 41:07
    STUDENT: What would that
    represent, geometrically,
  • 41:07 - 41:09
    the triple integral of x, y, z?
  • 41:09 - 41:12
    MAGDALENA TODA: It's a
    weighted triple integral.
  • 41:12 - 41:16
    I'm going to give
    you examples later.
  • 41:16 - 41:20
    When you have mass and momentum,
    when you compute the center
  • 41:20 - 41:25
    map, or you compute the
    mass, and somebody give you
  • 41:25 - 41:26
    densities.
  • 41:26 - 41:33
    Let me get -- If you have a
    triple integral over row at x,
  • 41:33 - 41:37
    y, z, this could be it,
    but I [? recall ?] it row
  • 41:37 - 41:40
    for a reason, not just for fun.
  • 41:40 - 41:43
    And here, dx, dy, dz.
  • 41:43 - 41:45
    Very good question, and
    it's very insightful.
  • 41:45 - 41:48
    For a physicist or
    engineer, the guy
  • 41:48 - 41:52
    needs to know why we take
    this weighted [? integral. ?]
  • 41:52 - 41:55
    If row is the
    density of an object,
  • 41:55 - 41:59
    if it's everywhere the same, if
    row is a homogeneous density,
  • 41:59 - 42:02
    for that piece of cheddar
    cheese-- Oh my God
  • 42:02 - 42:05
    I'm so hungry-- row
    would be constant.
  • 42:05 - 42:08
    If it's a quality cheddar
    made in Vermont in the best
  • 42:08 - 42:13
    factory, whatever, row would
    be considered to be a constant,
  • 42:13 - 42:14
    right?
  • 42:14 - 42:15
    And in that case, what happens?
  • 42:15 - 42:18
    If it's a constant,
    it's a gets out,
  • 42:18 - 42:21
    and then you have row
    times triple integral 1
  • 42:21 - 42:23
    dV, which is what?
  • 42:23 - 42:25
    The volume.
  • 42:25 - 42:28
    And then the volume times the
    density of the piece of cheese
  • 42:28 - 42:29
    will be?
  • 42:29 - 42:30
    STUDENT: [INAUDIBLE]
  • 42:30 - 42:33
    MAGDALENA TODA: The mass
    of the piece of cheese,
  • 42:33 - 42:37
    in kilograms, because I
    think in kilograms because I
  • 42:37 - 42:38
    can eat more.
  • 42:38 - 42:39
    OK?
  • 42:39 - 42:42
    Actually, no, I'm just kidding.
  • 42:42 - 42:47
    You guys have really-- I
    mean, 2 pounds and 1 kilogram
  • 42:47 - 42:48
    is not the same thin.
  • 42:48 - 42:50
    Can somebody tell me why?
  • 42:50 - 42:52
    I mean, you know it's not
    the same thing because,
  • 42:52 - 42:54
    the approximation.
  • 42:54 - 42:58
    But I'm claiming you cannot
    compare pounds with kilograms
  • 42:58 - 42:58
    at all.
  • 42:58 - 43:00
    STUDENT: Pounds is
    a measure of weight,
  • 43:00 - 43:02
    whereas kilograms is
    a measure of mass.
  • 43:02 - 43:03
    MAGDALENA TODA: Excellent.
  • 43:03 - 43:05
    Kilogram is a measure
    of mass, pound
  • 43:05 - 43:08
    is a measure of the
    gravitational force.
  • 43:08 - 43:11
    It's a force measure.
  • 43:11 - 43:16
    So OK.
  • 43:16 - 43:20
  • 43:20 - 43:23
    Which reminds me,
    there was-- I don't
  • 43:23 - 43:25
    know if you saw this short
    movie for 15 minutes that
  • 43:25 - 43:29
    got an award the
    previous Oscar last year,
  • 43:29 - 43:35
    and there was an old lady
    telling another old lady
  • 43:35 - 43:42
    in Great Britain, get
    2 pounds of sausage.
  • 43:42 - 43:45
    And the other one says,
    I thought we got metric,
  • 43:45 - 43:47
    because we are in
    the European Union.
  • 43:47 - 43:51
    And she said, then get me
    just the one meter of sausage,
  • 43:51 - 43:53
    or something.
  • 43:53 - 43:55
    So it was funny.
  • 43:55 - 43:56
    So it can be mass.
  • 43:56 - 44:00
    But what if this
    density is not the same?
  • 44:00 - 44:04
    This is exactly why we
    need to do the integral.
  • 44:04 - 44:08
    Imagine that the
    density is-- we have
  • 44:08 - 44:11
    a piece of cake with layers.
  • 44:11 - 44:13
    And again, you see
    how hungry I am.
  • 44:13 - 44:19
    So you have a layer, and
    then cream, or whipped cream,
  • 44:19 - 44:22
    or mousse, and another
    layer, and another mousse.
  • 44:22 - 44:25
    The density will vary.
  • 44:25 - 44:29
    But then there are bodies in
    physics where the density is
  • 44:29 - 44:31
    even a smooth function.
  • 44:31 - 44:38
    It doesn't matter that you have
    such a discontinuous function.
  • 44:38 - 44:39
    What would you do?
  • 44:39 - 44:40
    You just split.
  • 44:40 - 44:47
    You have triple row 1 for the
    first layer, then triple row 2
  • 44:47 - 44:50
    for the second later,
    the layer of mousse,
  • 44:50 - 44:53
    and then let's
    say it's tiramisu,
  • 44:53 - 44:58
    you have another layer, row
    three, dV3 for the top layer
  • 44:58 - 44:59
    of the tiramisu.
  • 44:59 - 45:00
    STUDENT: Can any row
    be kept constant?
  • 45:00 - 45:03
    MAGDALENA TODA: So
    these are discontinuous.
  • 45:03 - 45:05
    They are all constant, though.
  • 45:05 - 45:07
    That would be the
    great advantage,
  • 45:07 - 45:11
    because presumably mousse would
    have the constant density,
  • 45:11 - 45:15
    the dough has a constant,
    homogeneous density, and so on.
  • 45:15 - 45:19
    But what if the density
    varies in that body from point
  • 45:19 - 45:20
    to point?
  • 45:20 - 45:23
    Then nobody can do
    it by approximation.
  • 45:23 - 45:27
    You'd say volume, mass 1 plus
    mass 2 plus mass 3 plus mass 1.
  • 45:27 - 45:31
    You have to have a triple
    integral where this row varies,
  • 45:31 - 45:33
    constantly varies.
  • 45:33 - 45:35
    And for an engineer,
    that would be a puzzle.
  • 45:35 - 45:39
    Poor engineers says,
    oh my God, the density
  • 45:39 - 45:40
    is different from
    one point to another.
  • 45:40 - 45:43
    I have to find an
    approximated function
  • 45:43 - 45:48
    for that density moving from one
    point to another on that body.
  • 45:48 - 45:55
    And then the only way to do it
    would be to solve an integral.
  • 45:55 - 45:57
    Imagine that somebody--
    now it just occurred,
  • 45:57 - 46:02
    I never thought
    about it-- we would
  • 46:02 - 46:05
    be measured in terms of
    this type of integral.
  • 46:05 - 46:11
    Of course, people would be able
    to measure mass right away.
  • 46:11 - 46:13
    But then, if you were
    to know the density--
  • 46:13 - 46:18
    you cannot even know the density
    at every point of the body.
  • 46:18 - 46:21
    It varies a lot, so
    every point of our bodies
  • 46:21 - 46:26
    has a different
    material and a density.
  • 46:26 - 46:27
    OK.
  • 46:27 - 46:28
    STUDENT: Tiramisu. [INAUDIBLE]
  • 46:28 - 46:31
  • 46:31 - 46:32
    MAGDALENA TODA: Huh?
  • 46:32 - 46:33
    STUDENT: So you
    use the tiramasu,
  • 46:33 - 46:34
    you're making me hungry.
  • 46:34 - 46:35
    MAGDALENA TODA:
    Yeah, because now,
  • 46:35 - 46:38
    OK take your mind
    off the tiramisu.
  • 46:38 - 46:40
    Think about an exam.
  • 46:40 - 46:42
    Then you don't--
  • 46:42 - 46:45
    STUDENT: Now I'm sick.
  • 46:45 - 46:46
    MAGDALENA TODA: Exactly.
  • 46:46 - 46:50
    Now you need something
    against nausea.
  • 46:50 - 46:54
    Let's see what else
    is interesting to do.
  • 46:54 - 46:58
  • 46:58 - 47:00
    I'll give you ten minutes.
  • 47:00 - 47:02
    How much did I steal from you?
  • 47:02 - 47:07
    I stole constantly about
    five minutes of your breaks
  • 47:07 - 47:10
    for the last few Tuesdays.
  • 47:10 - 47:12
    STUDENT: So the integral--
  • 47:12 - 47:14
    MAGDALENA TODA: The
    integral of that.
  • 47:14 - 47:19
    I think I would be fair
    to give you 10 minutes
  • 47:19 - 47:22
    as a gift today to compensate.
  • 47:22 - 47:28
    OK, so remind me to let
    you go 10 minutes early.
  • 47:28 - 47:32
    Especially since
    spring break is coming.
  • 47:32 - 47:37
    We have a 3D application.
  • 47:37 - 47:40
    We have several 3D applications.
  • 47:40 - 47:44
    Let me see which one
    I want to mimic first.
  • 47:44 - 47:49
  • 47:49 - 47:51
    Yeah.
  • 47:51 - 47:56
    I'm going to pick my favorite,
    because I just want to.
  • 47:56 - 48:02
  • 48:02 - 48:16
    So imagine you
    have a disc that is
  • 48:16 - 48:19
    x squared plus y squared
    equals 1 would be the circle.
  • 48:19 - 48:22
    That's the unit
    disc on the floor.
  • 48:22 - 48:27
  • 48:27 - 48:38
    And then I have the plane
    x plus y plus z equals 8.
  • 48:38 - 48:40
    Then I'm going to
    draw that plane.
  • 48:40 - 48:41
    I'll try my best.
  • 48:41 - 48:49
  • 48:49 - 48:51
    It's similar to two
    examples from the book,
  • 48:51 - 48:54
    but I did not want to
    repeat the ones in the book
  • 48:54 - 48:57
    because I want you to
    actually read them.
  • 48:57 - 48:59
    That's kind of the idea.
  • 48:59 - 49:06
    So you have this
    picture, and you
  • 49:06 - 49:11
    realize that we had that
    in the first octant before.
  • 49:11 - 49:16
    So I say, I don't
    want the volume
  • 49:16 - 49:21
    of the body over the whole disc,
    only over the part of the disc
  • 49:21 - 49:24
    which is in the first octant.
  • 49:24 - 49:33
    So I say, I want this domain
    D, which is going to be what?
  • 49:33 - 49:36
    x squared plus y squared
    less than or equal to 1
  • 49:36 - 49:40
    in plane, with x
    positive, y positive.
  • 49:40 - 49:43
    Do you know what we call
    that in trigonometry?
  • 49:43 - 49:47
  • 49:47 - 49:50
    Does anybody know what we
    call this in trigonometry?
  • 49:50 - 49:58
  • 49:58 - 50:00
    Let me put the points
    while you think.
  • 50:00 - 50:03
    Hopefully, you are
    thinking about this.
  • 50:03 - 50:07
    This is 1 in x-axis.
  • 50:07 - 50:14
    1, 0, 0, and this is
    0, 1, 0, and this is y.
  • 50:14 - 50:18
    If I were to go up
    until I meet the plane,
  • 50:18 - 50:20
    what point would this--
  • 50:20 - 50:22
    STUDENT: [INAUDIBLE]
  • 50:22 - 50:27
    MAGDALENA TODA: What point
    would this-- on the thing.
  • 50:27 - 50:30
  • 50:30 - 50:35
    STUDENT: 1, 0, 7
    and then 0, 1, 7.
  • 50:35 - 50:36
    MAGDALENA TODA: 1, 0, 7.
  • 50:36 - 50:41
  • 50:41 - 50:47
    This would be you said 0, 1, 6.
  • 50:47 - 50:49
    And this would be 1, 0, 7.
  • 50:49 - 50:51
    How did you think about this?
  • 50:51 - 50:52
    How do you know?
  • 50:52 - 50:53
    STUDENT: Y plus z--
  • 50:53 - 50:57
    MAGDALENA TODA: Because z,
    because it's on the y-axis,
  • 50:57 - 51:01
    and since you are on the
    x-axis here, y has to be 0.
  • 51:01 - 51:02
    So you're right.
  • 51:02 - 51:02
    Very good.
  • 51:02 - 51:03
    Excellent.
  • 51:03 - 51:11
    Now I'm going to
    say, I'd like to know
  • 51:11 - 51:34
    the-- compute the volume of
    the body that is bounded above
  • 51:34 - 51:52
    from above by x plus y plus
    z equals 8, who's projection
  • 51:52 - 52:01
    on the floor is the
    domain D. And I'll say
  • 52:01 - 52:02
    volume of the cylindrical body.
  • 52:02 - 52:10
  • 52:10 - 52:15
    So how could you obtain
    such a, again-- No,
  • 52:15 - 52:17
    this is Murphy's Law.
  • 52:17 - 52:25
    OK, how could you obtain such
    an object, such a cylinder?
  • 52:25 - 52:27
    STUDENT: Take a pencil,
    and cut it into fourths.
  • 52:27 - 52:28
    MAGDALENA TODA: Huh?
  • 52:28 - 52:31
    STUDENT: Take like a cylindrical
    pencil and cut it into fourths.
  • 52:31 - 52:34
    MAGDALENA TODA: Take a
    salami, a piece of salami.
  • 52:34 - 52:40
    Cut that piece of salami into
    four, into four quarters.
  • 52:40 - 52:44
  • 52:44 - 52:50
    And then we take, we slice,
    and we slice like that.
  • 52:50 - 52:52
    So we have something like--
  • 52:52 - 52:54
    STUDENT: I tried to think
    of a non- food example.
  • 52:54 - 52:56
    MAGDALENA TODA: --a quarter.
  • 52:56 - 52:59
    How can I draw this?
  • 52:59 - 53:01
    OK, this is what it means.
  • 53:01 - 53:03
    You don't see this one.
  • 53:03 - 53:05
    You don't see this part.
  • 53:05 - 53:06
    You don't see this part.
  • 53:06 - 53:07
    This is curved.
  • 53:07 - 53:11
    And here, instead of cutting
    with another perpendicular
  • 53:11 - 53:13
    plane, along the
    salami-- so this
  • 53:13 - 53:18
    is the axis of the salami--
    instead of taking the knife
  • 53:18 - 53:21
    and cutting like that, I'm
    cutting an oblique plane,
  • 53:21 - 53:26
    and this is what this
    oblique plane will do.
  • 53:26 - 53:28
    STUDENT: If you cut
    that way, then you
  • 53:28 - 53:31
    would have only squares.
  • 53:31 - 53:32
    MAGDALENA TODA: Hmm?
  • 53:32 - 53:43
    So I'm going to have some
    oblique-- I cannot draw better.
  • 53:43 - 53:45
    I don't know how to draw better.
  • 53:45 - 53:47
    So it's going to be an
    oblique cut in the salami.
  • 53:47 - 53:50
  • 53:50 - 53:53
    Let's think how we
    do this problem.
  • 53:53 - 53:55
    Elementary, it will
    be a piece of cake--
  • 53:55 - 53:58
    it would be a piece of--
  • 53:58 - 53:59
    STUDENT: A piece of salami.
  • 53:59 - 54:00
    MAGDALENA TODA: No.
  • 54:00 - 54:01
    It wouldn't be apiece of salami.
  • 54:01 - 54:03
    STUDENT: It could be done.
  • 54:03 - 54:06
    MAGDALENA TODA: How could we do
    that quickly with the Calculus
  • 54:06 - 54:07
    III we know?
  • 54:07 - 54:09
    STUDENT: Find the
    triple integral.
  • 54:09 - 54:12
    Oh, you want us to do
    the double integral?
  • 54:12 - 54:14
    MAGDALENA TODA: Double, triple,
    I don't know what to do.
  • 54:14 - 54:16
    What do you think is best?
  • 54:16 - 54:18
    Let's do that triple
    integral first,
  • 54:18 - 54:21
    and you'll see that it's the
    same thing as double integral.
  • 54:21 - 54:31
    Triple integral over B, the
    body of the salami, 1 dV.
  • 54:31 - 54:34
    How can we set it up?
  • 54:34 - 54:37
    Well, this is a
    little bit tricky.
  • 54:37 - 54:39
    It's going to be like that.
  • 54:39 - 54:42
  • 54:42 - 54:45
    We can say, I have a double
    integral over my domain,
  • 54:45 - 54:52
    D. When it comes to the z,
    mister z has to be first.
  • 54:52 - 54:54
    So mister z says, I'm first.
  • 54:54 - 54:56
    I know where I'm going.
  • 54:56 - 55:00
    You guys, x and y
    are bound together,
  • 55:00 - 55:03
    mired in the element
    of area of the circles.
  • 55:03 - 55:06
    This is like dx dy.
  • 55:06 - 55:08
    But I am independent from you.
  • 55:08 - 55:09
    I am z.
  • 55:09 - 55:13
    So I'm going all the way
    from the floor to what?
  • 55:13 - 55:15
    You taught me that.
  • 55:15 - 55:20
    8 minus x minus y, and 1.
  • 55:20 - 55:22
    This is the way to do
    it as a triple integral,
  • 55:22 - 55:25
    but then Alex will
    say, I could have
  • 55:25 - 55:26
    done this as a double integral.
  • 55:26 - 55:29
    Let me show you how.
  • 55:29 - 55:33
    I could have done it over
    the domain D in plane.
  • 55:33 - 55:36
    Put the function,
    8 minus x minus y
  • 55:36 - 55:38
    is [? B and ?] z from
    the very beginning,
  • 55:38 - 55:43
    because that's my altitude
    function, f of x and y.
  • 55:43 - 55:47
    So then I say dx dy, dx
    dy, it doesn't matter.
  • 55:47 - 55:48
    That's the only theory element.
  • 55:48 - 55:49
    Fine.
  • 55:49 - 55:51
    It's the same thing.
  • 55:51 - 55:54
    This is what I wanted
    you to observe.
  • 55:54 - 55:56
    Whether you view it like the
    triple integral like that,
  • 55:56 - 55:58
    or you view it as the
    double integral like that,
  • 55:58 - 56:02
    it's the same thing.
  • 56:02 - 56:03
    This is not a headache.
  • 56:03 - 56:06
    The headache is coming next.
  • 56:06 - 56:08
    This is not a headache.
  • 56:08 - 56:11
    So you can do it in two ways.
  • 56:11 - 56:14
    And I'd like to
    look at the-- check
  • 56:14 - 56:20
    the two methods of doing this.
  • 56:20 - 56:24
  • 56:24 - 56:27
    And set up the integrals
    without solving them.
  • 56:27 - 56:38
  • 56:38 - 56:39
    Can you read my mind?
  • 56:39 - 56:42
    Do you realize what I'm asking?
  • 56:42 - 56:44
    Imagine that would
    be on the midterm.
  • 56:44 - 56:47
    What do you think I'm
    asking, the two methods?
  • 56:47 - 56:49
    This can be interpreted
    in many ways.
  • 56:49 - 56:50
    There are two methods.
  • 56:50 - 56:54
    I mean, one method by
    doing it with Cartesian
  • 56:54 - 56:56
    coordinates x and y.
  • 56:56 - 56:58
    The other method is switching
    to polar coordinates
  • 56:58 - 57:01
    and set up the integral
    without solving.
  • 57:01 - 57:04
    And you say, why not solving?
  • 57:04 - 57:05
    Because I'm going to cheat.
  • 57:05 - 57:08
    I'm going to use a
    TI-92 to solve it,
  • 57:08 - 57:11
    or I'm going to use
    a Matlab or Maple.
  • 57:11 - 57:13
    If it looks a little
    bit complicated,
  • 57:13 - 57:15
    then I don't want
    to spend my time.
  • 57:15 - 57:19
    Actually, engineers,
    after taking Calc III,
  • 57:19 - 57:20
    they know a lot.
  • 57:20 - 57:23
    They understand a lot
    about volumes, areas.
  • 57:23 - 57:27
    But do you think if you work on
    a real-life problem like that,
  • 57:27 - 57:29
    that your boss will
    let you waste your time
  • 57:29 - 57:31
    and do the integral by hand?
  • 57:31 - 57:31
    STUDENT: No.
  • 57:31 - 57:34
    MAGDALENA TODA: Most integrals
    are really complicated
  • 57:34 - 57:35
    in everyday life.
  • 57:35 - 57:36
    So what you're
    going to do is going
  • 57:36 - 57:40
    to be a scientific software,
    like Matlab, which is primarily
  • 57:40 - 57:44
    for engineers, Mathematica,
    which is similar to Matlab,
  • 57:44 - 57:46
    but is mainly for
    mathematicians.
  • 57:46 - 57:48
    It was invented
    at the University
  • 57:48 - 57:51
    of Illinois Urbana-Champaign,
    and they're still
  • 57:51 - 57:52
    very proud of it.
  • 57:52 - 57:54
    I prefer Matlab
    because I feel Matlab
  • 57:54 - 57:58
    is stronger, has higher
    capabilities than Mathematica.
  • 57:58 - 57:59
    You can use Maple.
  • 57:59 - 58:05
    Maple lets you set up the
    endpoints even as functions.
  • 58:05 - 58:08
    And then it's user
    friendly, you type in this,
  • 58:08 - 58:10
    you type in the endpoints.
  • 58:10 - 58:12
    It has little windows, here.
  • 58:12 - 58:14
    You don't need to
    know any programming.
  • 58:14 - 58:18
    It's made for people who
    have no programming skills.
  • 58:18 - 58:20
    So it's going to show
    a little window on top,
  • 58:20 - 58:22
    here, here, here and here.
  • 58:22 - 58:24
    You [? have ?] those,
    and you press Enter,
  • 58:24 - 58:27
    and it's going to spit
    the answer back at you.
  • 58:27 - 58:29
    So this is how
    engineers actually
  • 58:29 - 58:31
    solve the everyday integrals.
  • 58:31 - 58:33
    Not by hand.
  • 58:33 - 58:36
    I want to be able to
    set it up in both ways
  • 58:36 - 58:39
    before I go home or
    eat something, right?
  • 58:39 - 58:44
    So we don't have to spend
    a lot of time on it.
  • 58:44 - 58:48
    But if you want to tell me
    how I am going to set it up,
  • 58:48 - 58:50
    I would be very grateful.
  • 58:50 - 58:54
    So this is Cartesian,
    and this is polar.
  • 58:54 - 59:06
  • 59:06 - 59:07
    All right.
  • 59:07 - 59:08
    Who helps me?
  • 59:08 - 59:10
    In Cartesian-- which
    one do you prefer?
  • 59:10 - 59:11
    I mean, it doesn't matter.
  • 59:11 - 59:14
    You guys are good
    and smart, and you'll
  • 59:14 - 59:16
    figure out what I need to do.
  • 59:16 - 59:19
    If I want to do it in terms
    of vertical strip-- so
  • 59:19 - 59:21
    for vertical strip
    method-- first
  • 59:21 - 59:25
    I integrate with respect to
    y, and then with respect to x.
  • 59:25 - 59:27
    And maybe, to test
    your understanding,
  • 59:27 - 59:30
    let me change the
    order of integrals
  • 59:30 - 59:33
    and see how much you
    understood from that last time.
  • 59:33 - 59:35
    STUDENT: [INAUDIBLE]
  • 59:35 - 59:39
    MAGDALENA TODA: So x is
    between what and what?
  • 59:39 - 59:40
    STUDENT: 0 and 1.
  • 59:40 - 59:41
    MAGDALENA TODA: Look
    at this picture.
  • 59:41 - 59:43
    I have to reproduce
    this picture like that.
  • 59:43 - 59:47
    0 to 1, says Alex,
    and he's right.
  • 59:47 - 59:49
    And why will he decide against--
  • 59:49 - 59:50
    STUDENT: 1 minus x squared.
  • 59:50 - 59:53
    MAGDALENA TODA: --square
    root 1 minus x squared.
  • 59:53 - 59:55
    So we know very well
    what we are going to do,
  • 59:55 - 59:57
    what Maple is
    going to do for us.
  • 59:57 - 60:00
    1 square root 1 minus x squared.
  • 60:00 - 60:02
    And then what do I put here?
  • 60:02 - 60:03
    8 minus x minus y.
  • 60:03 - 60:06
    Can I do it by hand?
  • 60:06 - 60:08
    Yes, I guarantee to you
    I can do it by hand.
  • 60:08 - 60:10
    Let me tell you why.
  • 60:10 - 60:14
    Because when we integrate
    with respect to y, I get xy.
  • 60:14 - 60:18
    So I get xy, and y will be
    plugged in 1 minus x squared.
  • 60:18 - 60:24
    How am I going to solve
    an integral like this?
  • 60:24 - 60:28
    I can the first one with a
    table, the second one with a u
  • 60:28 - 60:30
    substitution.
  • 60:30 - 60:33
    On the last one is a
    little bit painful.
  • 60:33 - 60:35
    I'm going to have
    y squared over 2--
  • 60:35 - 60:36
    STUDENT: That's
    the easiest part.
  • 60:36 - 60:39
    MAGDALENA TODA: According
    to Alex, yes, you're right.
  • 60:39 - 60:41
    Maybe that is the easiest.
  • 60:41 - 60:43
    STUDENT: That's the [INAUDIBLE]
    part you can integrate--
  • 60:43 - 60:45
    MAGDALENA TODA: And I can
    integrate one at a time,
  • 60:45 - 60:47
    and I'm going to
    waste all my time.
  • 60:47 - 60:49
    So if I want to be an
    efficient engineer,
  • 60:49 - 60:54
    and my boss is waiting for
    the end-of-the-day project,
  • 60:54 - 60:56
    of course I'm not going
    to do this by hand.
  • 60:56 - 60:59
    How about the other integral?
  • 60:59 - 61:02
    Same integral.
  • 61:02 - 61:05
    Same idea, y between 0 and 1.
  • 61:05 - 61:07
    And x between 0 and
  • 61:07 - 61:08
    STUDENT: Square root
    of 1 minus y squared.
  • 61:08 - 61:11
    MAGDALENA TODA: Square
    root of 1 minus y squared.
  • 61:11 - 61:16
    Because I'll do this guy
    with horizontal strips,
  • 61:16 - 61:19
    and forget about
    the vertical strips.
  • 61:19 - 61:23
    And here's the y-- I rotate
    my head and it cracks,
  • 61:23 - 61:28
    so that means that
    I need some yoga.
  • 61:28 - 61:31
    y is between 0 and 1.
  • 61:31 - 61:31
    Or gymnastics.
  • 61:31 - 61:36
    So x is between
    0 and square root
  • 61:36 - 61:40
    1 minus y squared. [INAUDIBLE].
  • 61:40 - 61:42
    And I'll leave it
    here on the meter.
  • 61:42 - 61:46
    And I'm going to make a
    sample like I promised.
  • 61:46 - 61:49
    OK, good.
  • 61:49 - 61:53
    How would you do this to set up
    the polar coordinate integral?
  • 61:53 - 61:58
    And that is why Alex said
    maybe that's a pain because
  • 61:58 - 62:00
    of a reason.
  • 62:00 - 62:04
    And he's right, it's a little
    bit painful to solve by hand.
  • 62:04 - 62:07
    But again, once you
    switch to polar,
  • 62:07 - 62:11
    you can solve it with a
    calculator or a computer
  • 62:11 - 62:14
    software, scientific
    software in no time.
  • 62:14 - 62:19
    In Maple, you just have
    to plug in the numbers.
  • 62:19 - 62:22
    You cannot plug in theta,
    I think, as a symbol.
  • 62:22 - 62:23
    I'm not sure.
  • 62:23 - 62:27
    But you can put theta
    as t and r will be r,
  • 62:27 - 62:29
    or you can use
    whatever letters you
  • 62:29 - 62:31
    want that are roman letters.
  • 62:31 - 62:35
    So you have to
    integrate smartly, here,
  • 62:35 - 62:38
    switching to r and
    theta, and think
  • 62:38 - 62:41
    about the meaning of that.
  • 62:41 - 62:45
    So first of all, if
    I put dr d theta,
  • 62:45 - 62:49
    I'm not worried that you won't
    be able to get r and theta,
  • 62:49 - 62:51
    because I know you can do it.
  • 62:51 - 62:56
    You can prove it to me
    right now. r between 0 and
  • 62:56 - 62:57
    STUDENT: 1.
  • 62:57 - 62:58
    MAGDALENA TODA: Excellent.
  • 62:58 - 63:01
    And theta, pay
    attention, between 0 and
  • 63:01 - 63:02
    STUDENT: pi over 2
  • 63:02 - 63:03
    MAGDALENA TODA: Excellent.
  • 63:03 - 63:04
    I'm proud.
  • 63:04 - 63:04
    Yes, sir?
  • 63:04 - 63:07
    STUDENT: Is it supposed
    to be r dr 2 theta,
  • 63:07 - 63:08
    or are you going
    to add that later?
  • 63:08 - 63:10
    MAGDALENA TODA: I
    will add it here.
  • 63:10 - 63:13
    So the integrand
    will contain the r.
  • 63:13 - 63:17
    Now what do I put
    in terms of this?
  • 63:17 - 63:19
    I left enough room.
  • 63:19 - 63:23
    STUDENT: Is it pi over 2,
    or is it negative pi over 2?
  • 63:23 - 63:25
    MAGDALENA TODA:
    It doesn't matter,
  • 63:25 - 63:31
    because I'll have to take that--
    we assume always theta to go
  • 63:31 - 63:35
    counterclockwise, and go
    between 0 and pi over 2,
  • 63:35 - 63:39
    so that when you start--
    let me make this motion.
  • 63:39 - 63:41
    You are here at theta equals 0.
  • 63:41 - 63:42
    STUDENT: Oh, OK.
  • 63:42 - 63:42
    Sorry.
  • 63:42 - 63:44
    I got my coordinates
    mixed around--
  • 63:44 - 63:46
    MAGDALENA TODA: --and
    counterclockwise to pi over 2.
  • 63:46 - 63:47
    [INTERPOSING VOICES]
  • 63:47 - 63:49
  • 63:49 - 63:50
    MAGDALENA TODA: Yeah.
  • 63:50 - 63:55
    So you go in the trigonometric--
    Here, you have 8 minus,
  • 63:55 - 63:58
    and who tells me what
    I'm supposed to type?
  • 63:58 - 63:59
    STUDENT: r over x.
  • 63:59 - 64:08
    MAGDALENA TODA: r
    cosine theta minus
  • 64:08 - 64:09
    STUDENT: Sine.
  • 64:09 - 64:11
    MAGDALENA TODA: r sine theta.
  • 64:11 - 64:14
    And let mister
    whatever his name is,
  • 64:14 - 64:17
    the computer, find the answer.
  • 64:17 - 64:19
    Can I do it by hand?
  • 64:19 - 64:21
    Actually, I can.
  • 64:21 - 64:27
    I can, but again, it's not worth
    it, because it drives me crazy.
  • 64:27 - 64:30
    How would I do it by hand?
  • 64:30 - 64:32
    I would split the
    integral into three,
  • 64:32 - 64:36
    and I would easily
    compute 8 times r,
  • 64:36 - 64:37
    integrand is going to be easy.
  • 64:37 - 64:38
    Right?
  • 64:38 - 64:39
    Agree with me?
  • 64:39 - 64:41
    Then what am I going to do?
  • 64:41 - 64:47
    I'm going to say, an r out
    times an r, out comes r squared.
  • 64:47 - 64:51
    And I have integral of r
    squared times a function
  • 64:51 - 64:54
    of theta only,
    which is going to be
  • 64:54 - 64:56
    sine theta plus cosine theta.
  • 64:56 - 64:59
    We are going to say, yes,
    with a minus, with a minus.
  • 64:59 - 65:02
  • 65:02 - 65:07
    Now, when I compute
    r and theta thingy,
  • 65:07 - 65:10
    theta will be between
    0 and pi over 2.
  • 65:10 - 65:12
    r will be between 0 and 1.
  • 65:12 - 65:16
    But I don't care, because
    Matthew reminded me,
  • 65:16 - 65:19
    if you have a product
    of separate variables,
  • 65:19 - 65:22
    life becomes all of the
    sudden easier for you.
  • 65:22 - 65:25
    STUDENT: You've also got to
    add your integral of [? 8r ?]
  • 65:25 - 65:25
    [? dr. ?]
  • 65:25 - 65:26
    MAGDALENA TODA: Yeah.
  • 65:26 - 65:28
    At the end, I'm going to
    add the integral of 8r.
  • 65:28 - 65:30
    So I take them separately.
  • 65:30 - 65:33
    I just look at one chunk.
  • 65:33 - 65:35
    And this chunk will be what?
  • 65:35 - 65:39
    Can you even see how easy it's
    going to be with the naked eye?
  • 65:39 - 65:41
    Firs of all,
    integral from 0 to 1,
  • 65:41 - 65:44
    r squared dr is a piece of cake.
  • 65:44 - 65:46
    How much is that--
    piece of salami.
  • 65:46 - 65:47
    STUDENT: 1/3.
  • 65:47 - 65:49
    MAGDALENA TODA: 1/3.
  • 65:49 - 65:50
    Right?
  • 65:50 - 65:52
    Because it's r cubed over 3.
  • 65:52 - 65:53
    Then you have 1/3.
  • 65:53 - 65:55
    That's easy.
  • 65:55 - 65:58
    With a minus in front, but I
    don't care about it in the end.
  • 65:58 - 66:04
    What is the integral of
    sine theta cosine theta?
  • 66:04 - 66:06
    STUDENT: Negative [INAUDIBLE].
  • 66:06 - 66:11
    MAGDALENA TODA: Minus
    cosine theta plus sine theta
  • 66:11 - 66:14
    taken between 0 and pi over 2.
  • 66:14 - 66:16
    Will this be hard?
  • 66:16 - 66:20
    Who's going to tell me what,
    or how I'm going to get what--
  • 66:20 - 66:23
    we don't compute it now,
    but I just give you.
  • 66:23 - 66:24
    Cosine of pi over 3 is?
  • 66:24 - 66:25
    STUDENT: 0.
  • 66:25 - 66:25
    MAGDALENA TODA: 0.
  • 66:25 - 66:27
    Sine of pi over 2 is?
  • 66:27 - 66:28
    STUDENT: Oh yeah.
  • 66:28 - 66:28
    1.
  • 66:28 - 66:28
    MAGDALENA TODA: 1.
  • 66:28 - 66:31
    So this is going to
    be 1 minus, what's
  • 66:31 - 66:35
    the whole thingy computed at 0?
  • 66:35 - 66:35
    STUDENT: [INAUDIBLE].
  • 66:35 - 66:39
    MAGDALENA TODA: It's going to
    be minus 1, but minus minus 1
  • 66:39 - 66:40
    is plus 1.
  • 66:40 - 66:43
    So I have 2.
  • 66:43 - 66:46
    So only this chunk of the
    integral would be easy.
  • 66:46 - 66:47
    Minus 2/3.
  • 66:47 - 66:48
    OK?
  • 66:48 - 66:51
    So it can be done by hand,
    but why waste the time when
  • 66:51 - 66:53
    you can do it with Maple?
  • 66:53 - 66:53
    Yes, sir?
  • 66:53 - 66:56
    STUDENT: Where did
    you get rid of 8?
  • 66:56 - 66:58
    On the second, after the 8--
  • 66:58 - 67:00
    MAGDALENA TODA: No, I didn't.
  • 67:00 - 67:02
    That's exactly what
    we were talking.
  • 67:02 - 67:06
    Alex says, but you just
    talked about integral of 8r,
  • 67:06 - 67:07
    but you didn't want to do it.
  • 67:07 - 67:09
    I said, I didn't want to do it.
  • 67:09 - 67:13
    This is just the second
    chunk of this integral.
  • 67:13 - 67:17
    So I know that I can do integral
    of integral of 8r in no time.
  • 67:17 - 67:20
    Then I would need to
    take this and add that,
  • 67:20 - 67:21
    and get the number.
  • 67:21 - 67:23
    I don't care about the number.
  • 67:23 - 67:25
    I just care about the method.
  • 67:25 - 67:25
    Yes, sir?
  • 67:25 - 67:28
    STUDENT: Why are the limits
    from 0 to 1 instead of like 0
  • 67:28 - 67:30
    to r squared?
  • 67:30 - 67:33
    Because didn't we say
    earlier the domain
  • 67:33 - 67:35
    is x squared plus y squared?
  • 67:35 - 67:38
    Wouldn't that be r squared?
  • 67:38 - 67:38
    MAGDALENA TODA: No.
  • 67:38 - 67:39
    No, wait.
  • 67:39 - 67:40
    This is r squared.
  • 67:40 - 67:41
    STUDENT: Right.
  • 67:41 - 67:44
    Why didn't we plug r
    squared into the 1 again.
  • 67:44 - 67:47
    MAGDALENA TODA: And that means
    r is between 0 and 1, right?
  • 67:47 - 67:48
    STUDENT: Oh, OK.
  • 67:48 - 67:50
    MAGDALENA TODA: r squared
    being less than 1.
  • 67:50 - 67:52
    That means r is between 0 and 1.
  • 67:52 - 67:53
    OK?
  • 67:53 - 67:57
    And one last problem-- no.
  • 67:57 - 67:59
    No last problem.
  • 67:59 - 68:01
    We have barely 10 minutes.
  • 68:01 - 68:04
    So you read from the book some.
  • 68:04 - 68:08
    I will come back to this
    section, and I'll do review.
  • 68:08 - 68:11
    Have a wonderful
    spring break, and I'm
  • 68:11 - 68:14
    going to see you after
    spring break on Tuesday.
  • 68:14 - 68:16
    [INTERPOSING VOICES]
  • 68:16 - 68:18
Title:
TTU Math2450 Calculus3 Sec 12.5
Description:

Triple Integrals

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Video Language:
English

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