[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.41,0:00:05.56,Default,,0000,0000,0000,,In this video, we're going to be\Nhaving a look at how we find
Dialogue: 0,0:00:05.56,0:00:09.61,Default,,0000,0000,0000,,what's called the volume of a\Nsolid of revolution. So let's
Dialogue: 0,0:00:09.61,0:00:11.08,Default,,0000,0000,0000,,describe first of all.
Dialogue: 0,0:00:11.94,0:00:14.57,Default,,0000,0000,0000,,Solid of Revolution is.
Dialogue: 0,0:00:16.68,0:00:18.84,Default,,0000,0000,0000,,Suppose we've got a graph of a
Dialogue: 0,0:00:18.84,0:00:24.81,Default,,0000,0000,0000,,function. Why?\NEquals FX.
Dialogue: 0,0:00:26.00,0:00:29.62,Default,,0000,0000,0000,,So there's our graph, let's pick
Dialogue: 0,0:00:29.62,0:00:35.13,Default,,0000,0000,0000,,out. Two values of X, let's\Nsay X equals a.
Dialogue: 0,0:00:37.38,0:00:43.89,Default,,0000,0000,0000,,X equals B and we've identified\Npoints on the curve.
Dialogue: 0,0:00:45.80,0:00:52.52,Default,,0000,0000,0000,,Now let's imagine that we\Nrotate this section of the
Dialogue: 0,0:00:52.52,0:00:57.90,Default,,0000,0000,0000,,curve about the X axis\Nthrough 360 degrees.
Dialogue: 0,0:00:58.49,0:01:01.12,Default,,0000,0000,0000,,So this will come round in a
Dialogue: 0,0:01:01.12,0:01:05.14,Default,,0000,0000,0000,,small circle. And this will come\Nround in a big circle.
Dialogue: 0,0:01:05.67,0:01:11.31,Default,,0000,0000,0000,,And it will form a solid and we\Ncan sketch it. Very roughly,
Dialogue: 0,0:01:11.31,0:01:13.05,Default,,0000,0000,0000,,there's the front bit.
Dialogue: 0,0:01:13.61,0:01:18.89,Default,,0000,0000,0000,,And then stay. And then we'll\Nhave the curve looking like
Dialogue: 0,0:01:18.89,0:01:24.28,Default,,0000,0000,0000,,that. So we've got this curious\Nsort of Vars shape almost that
Dialogue: 0,0:01:24.28,0:01:26.66,Default,,0000,0000,0000,,we formed by rotating the curve.
Dialogue: 0,0:01:27.33,0:01:32.19,Default,,0000,0000,0000,,Now, given an appropriate curve,\Nlet's say it was a semi circle,
Dialogue: 0,0:01:32.19,0:01:37.05,Default,,0000,0000,0000,,let me just draw that curve,\Nsupposing we had a semi circle
Dialogue: 0,0:01:37.05,0:01:42.72,Default,,0000,0000,0000,,like that of radius R. So it\Nwould cross the X axis at minus
Dialogue: 0,0:01:42.72,0:01:48.39,Default,,0000,0000,0000,,R&R and the Y axis are. Now if\Nwe rotated that, we'd get a
Dialogue: 0,0:01:48.39,0:01:54.06,Default,,0000,0000,0000,,sphere. If we got the sphere and\Nwe had a way of calculating the
Dialogue: 0,0:01:54.06,0:01:58.92,Default,,0000,0000,0000,,volume, then we would know what\Nthe volume of a sphere was.
Dialogue: 0,0:01:59.00,0:02:03.39,Default,,0000,0000,0000,,Well, most people, most of you\Nwatching this probably already
Dialogue: 0,0:02:03.39,0:02:09.54,Default,,0000,0000,0000,,know that the volume is 4 thirds\Nπ R cubed. The question is where
Dialogue: 0,0:02:09.54,0:02:14.36,Default,,0000,0000,0000,,does that come from? How can we\Nactually make that calculation?
Dialogue: 0,0:02:15.25,0:02:21.21,Default,,0000,0000,0000,,Similarly, if we had a straight\Nline that went through the
Dialogue: 0,0:02:21.21,0:02:26.62,Default,,0000,0000,0000,,origin. So if we were to rotate\Nthat through 360 degrees then
Dialogue: 0,0:02:26.62,0:02:28.39,Default,,0000,0000,0000,,that would give us.
Dialogue: 0,0:02:29.15,0:02:33.33,Default,,0000,0000,0000,,A comb, so there's the end,\Ngiving us the cone, and that's
Dialogue: 0,0:02:33.33,0:02:35.41,Default,,0000,0000,0000,,the reflection in the X axis.
Dialogue: 0,0:02:36.18,0:02:43.00,Default,,0000,0000,0000,,And the volume of a cone we\Nknow is 1/3 Pi R-squared H.
Dialogue: 0,0:02:43.00,0:02:50.36,Default,,0000,0000,0000,,Where are is the radius of the\Nbase and H is the height of
Dialogue: 0,0:02:50.36,0:02:56.49,Default,,0000,0000,0000,,the cone. So again, where does\Nthis volume come from? How can
Dialogue: 0,0:02:56.49,0:03:01.41,Default,,0000,0000,0000,,we actually calculate that? So\Nthat's what we're going to be
Dialogue: 0,0:03:01.41,0:03:07.22,Default,,0000,0000,0000,,having a look at. How can we\Nfind these volumes of the solids
Dialogue: 0,0:03:07.22,0:03:12.58,Default,,0000,0000,0000,,that we get when we rotate\Ncurves about the X axis through
Dialogue: 0,0:03:12.58,0:03:19.44,Default,,0000,0000,0000,,360 degrees? So let's start off\Nmore or less the same diagram
Dialogue: 0,0:03:19.44,0:03:22.40,Default,,0000,0000,0000,,that I began with before.
Dialogue: 0,0:03:23.80,0:03:25.34,Default,,0000,0000,0000,,Section of the curve.
Dialogue: 0,0:03:25.99,0:03:28.39,Default,,0000,0000,0000,,Identify two points.
Dialogue: 0,0:03:29.18,0:03:34.58,Default,,0000,0000,0000,,On the X axis, X equals a,\NX equals B.
Dialogue: 0,0:03:35.55,0:03:41.63,Default,,0000,0000,0000,,Drawing the ordinance and will\Nsay we're going to rotate this
Dialogue: 0,0:03:41.63,0:03:47.72,Default,,0000,0000,0000,,piece of curve between these two\Nlimits through 360 degrees. Now
Dialogue: 0,0:03:47.72,0:03:54.35,Default,,0000,0000,0000,,let's have a look at what\Nhappens at a general point on
Dialogue: 0,0:03:54.35,0:03:58.22,Default,,0000,0000,0000,,the curve at X on the X
Dialogue: 0,0:03:58.22,0:04:03.94,Default,,0000,0000,0000,,axis. And when we rotate that,\Nthis ordinate will come round
Dialogue: 0,0:04:03.94,0:04:05.77,Default,,0000,0000,0000,,and make a circle.
Dialogue: 0,0:04:06.33,0:04:14.09,Default,,0000,0000,0000,,Let's move on a small\Ndistance Delta XA small positive
Dialogue: 0,0:04:14.09,0:04:16.42,Default,,0000,0000,0000,,increase in X.
Dialogue: 0,0:04:19.94,0:04:24.40,Default,,0000,0000,0000,,And let's again identify the\Nordinate going up to the curve.
Dialogue: 0,0:04:25.02,0:04:29.71,Default,,0000,0000,0000,,Now when we rotate these two\Ntogether, there isn't a great
Dialogue: 0,0:04:29.71,0:04:34.39,Default,,0000,0000,0000,,deal of difference between this\Nand this. Yes, we could say,
Dialogue: 0,0:04:34.39,0:04:39.93,Default,,0000,0000,0000,,well, This is why because this\Nis the point on the curve Y
Dialogue: 0,0:04:39.93,0:04:45.89,Default,,0000,0000,0000,,equals F of X and this is Y Plus\NDelta Y. The increase that
Dialogue: 0,0:04:45.89,0:04:47.60,Default,,0000,0000,0000,,results from the increasing
Dialogue: 0,0:04:47.60,0:04:54.59,Default,,0000,0000,0000,,Delta X. We spend this round\Nand we've almost got almost got
Dialogue: 0,0:04:54.59,0:04:56.19,Default,,0000,0000,0000,,a small disk.
Dialogue: 0,0:04:56.92,0:05:02.26,Default,,0000,0000,0000,,Small disk whose volume we can\Ncalculate if we think of it as
Dialogue: 0,0:05:02.26,0:05:06.37,Default,,0000,0000,0000,,being a cylinder. If we think\Nthere's so little difference
Dialogue: 0,0:05:06.37,0:05:10.89,Default,,0000,0000,0000,,between Y&Y Plus Delta Y that\Neffectively we can say all
Dialogue: 0,0:05:10.89,0:05:16.24,Default,,0000,0000,0000,,right. Let's say this is a disk\Nas it comes round. It's a
Dialogue: 0,0:05:16.24,0:05:20.76,Default,,0000,0000,0000,,circular disk. It's a cylinder.\NIn other words, and that the
Dialogue: 0,0:05:20.76,0:05:25.69,Default,,0000,0000,0000,,volume of this cylinder will be\NDelta V equals. Now the formula
Dialogue: 0,0:05:25.69,0:05:28.16,Default,,0000,0000,0000,,for the volume of a cylinder.
Dialogue: 0,0:05:28.23,0:05:33.68,Default,,0000,0000,0000,,Is that it's Pi R-squared H\Nlet me put that in inverted
Dialogue: 0,0:05:33.68,0:05:39.58,Default,,0000,0000,0000,,commas just to show that it is\Nthe formula for a cylinder and
Dialogue: 0,0:05:39.58,0:05:45.94,Default,,0000,0000,0000,,not quite what we've got here.\NSo this is going to be π and
Dialogue: 0,0:05:45.94,0:05:51.38,Default,,0000,0000,0000,,the radius of this disk is\Nwhy? So that's Pi Y squared
Dialogue: 0,0:05:51.38,0:05:55.02,Default,,0000,0000,0000,,and its height. Is this\Nthickness Delta X.
Dialogue: 0,0:05:56.18,0:06:02.94,Default,,0000,0000,0000,,So if we want the volume of the\Nwhole of the solid, we need to
Dialogue: 0,0:06:02.94,0:06:06.10,Default,,0000,0000,0000,,add up all of these little bits.
Dialogue: 0,0:06:06.11,0:06:12.31,Default,,0000,0000,0000,,So we want to add up all of\Nthose little bits of volume, and
Dialogue: 0,0:06:12.31,0:06:18.96,Default,,0000,0000,0000,,we want to add them up from X\Nequals a through two X equals B.
Dialogue: 0,0:06:19.64,0:06:22.91,Default,,0000,0000,0000,,So this will be.
Dialogue: 0,0:06:22.91,0:06:30.41,Default,,0000,0000,0000,,X equals a, X equals B and\Nwe want to put this in instead
Dialogue: 0,0:06:30.41,0:06:32.02,Default,,0000,0000,0000,,of Delta be.
Dialogue: 0,0:06:32.64,0:06:37.69,Default,,0000,0000,0000,,So that's Pi Y\Nsquared Delta X.
Dialogue: 0,0:06:38.89,0:06:44.07,Default,,0000,0000,0000,,Now what we need is a process\Nwhereby we don't actually have
Dialogue: 0,0:06:44.07,0:06:48.83,Default,,0000,0000,0000,,to do this summation, and in the\Nvideo integration of summation
Dialogue: 0,0:06:48.83,0:06:54.87,Default,,0000,0000,0000,,what we saw is that when we\Nhave, uh, some like this, if we
Dialogue: 0,0:06:54.87,0:07:00.06,Default,,0000,0000,0000,,take the limit as this Delta X,\Nthis small positive increment in
Dialogue: 0,0:07:00.06,0:07:05.67,Default,,0000,0000,0000,,X tends to 0. If we take that\Nlimit then this becomes an
Dialogue: 0,0:07:05.67,0:07:09.68,Default,,0000,0000,0000,,integral. So therefore I'm going\Nto take the limit.
Dialogue: 0,0:07:10.32,0:07:17.26,Default,,0000,0000,0000,,Let's wants down. We want\Nthe is equal to the
Dialogue: 0,0:07:17.26,0:07:17.95,Default,,0000,0000,0000,,limit.
Dialogue: 0,0:07:18.98,0:07:26.65,Default,,0000,0000,0000,,Ask Delta X tends to zero\Nof this. Some X equals A2,
Dialogue: 0,0:07:26.65,0:07:33.68,Default,,0000,0000,0000,,X equals B of Pi Y\Nsquared Delta X, and the
Dialogue: 0,0:07:33.68,0:07:41.34,Default,,0000,0000,0000,,notation that we have for this\Nas we see in the video
Dialogue: 0,0:07:41.34,0:07:48.37,Default,,0000,0000,0000,,integration as summation is the\Nintegral from A to B of
Dialogue: 0,0:07:48.37,0:07:51.89,Default,,0000,0000,0000,,Π. Y squared X.
Dialogue: 0,0:07:53.18,0:07:59.30,Default,,0000,0000,0000,,And that's the formula that will\Ngive us the volume of the solid
Dialogue: 0,0:07:59.30,0:08:04.96,Default,,0000,0000,0000,,of revolution when we rotate the\Ncurve Y equals F of X.
Dialogue: 0,0:08:05.74,0:08:09.23,Default,,0000,0000,0000,,That's contained between\Nthe ordinance X equals A
Dialogue: 0,0:08:09.23,0:08:10.97,Default,,0000,0000,0000,,and X equals B.
Dialogue: 0,0:08:12.01,0:08:14.37,Default,,0000,0000,0000,,Now let's use this.
Dialogue: 0,0:08:15.02,0:08:16.89,Default,,0000,0000,0000,,Formula in an example.
Dialogue: 0,0:08:17.41,0:08:19.37,Default,,0000,0000,0000,,An example I'm going to take is
Dialogue: 0,0:08:19.37,0:08:22.62,Default,,0000,0000,0000,,the sphere. In other words,\Nwe're going to rotate.
Dialogue: 0,0:08:23.21,0:08:27.14,Default,,0000,0000,0000,,A semi circle about the X axis.
Dialogue: 0,0:08:28.14,0:08:35.37,Default,,0000,0000,0000,,So I'll draw the semicircle 1st\Nand then put in the X&Y Axis,
Dialogue: 0,0:08:35.37,0:08:37.04,Default,,0000,0000,0000,,so there's Y.
Dialogue: 0,0:08:38.85,0:08:39.87,Default,,0000,0000,0000,,X.
Dialogue: 0,0:08:41.05,0:08:48.10,Default,,0000,0000,0000,,So semicircle so its radius will\Ntake to be R. And what's the
Dialogue: 0,0:08:48.10,0:08:55.14,Default,,0000,0000,0000,,equation of that curve that will\Nbe Y equals the square root of?
Dialogue: 0,0:08:55.68,0:08:59.08,Default,,0000,0000,0000,,R-squared minus X squared.
Dialogue: 0,0:09:00.24,0:09:04.12,Default,,0000,0000,0000,,So. This is our equation for Y.
Dialogue: 0,0:09:04.67,0:09:11.74,Default,,0000,0000,0000,,The volume of the sphere that\Nwe get when we rotate this
Dialogue: 0,0:09:11.74,0:09:17.63,Default,,0000,0000,0000,,semicircle through 360 degrees\Nis the integral from A to
Dialogue: 0,0:09:17.63,0:09:21.16,Default,,0000,0000,0000,,B of Pi Y squared DX.
Dialogue: 0,0:09:22.30,0:09:28.31,Default,,0000,0000,0000,,Now let's identify what the A\Nand the B and the Y squared are.
Dialogue: 0,0:09:28.31,0:09:30.88,Default,,0000,0000,0000,,So the A is minus R.
Dialogue: 0,0:09:32.20,0:09:34.63,Default,,0000,0000,0000,,The B is our.
Dialogue: 0,0:09:36.37,0:09:41.19,Default,,0000,0000,0000,,Pie Weiss Quetz. We\Nneed to square this.
Dialogue: 0,0:09:43.66,0:09:49.23,Default,,0000,0000,0000,,So that's the square root of\NR-squared minus X squared, and
Dialogue: 0,0:09:49.23,0:09:54.29,Default,,0000,0000,0000,,we need to square it integrated\Nwith respect to X.
Dialogue: 0,0:09:55.04,0:10:02.03,Default,,0000,0000,0000,,So we take the integral between\Nminus R&R of Pi. We do this
Dialogue: 0,0:10:02.03,0:10:07.41,Default,,0000,0000,0000,,squaring and that's our squared\Nminus X squared integrated with
Dialogue: 0,0:10:07.41,0:10:14.95,Default,,0000,0000,0000,,respect to X. So now we need\Nto look at this integral and we
Dialogue: 0,0:10:14.95,0:10:17.64,Default,,0000,0000,0000,,need to work it out.
Dialogue: 0,0:10:17.82,0:10:23.31,Default,,0000,0000,0000,,Turn the page and write it down\Nagain at the top of the page.
Dialogue: 0,0:10:23.92,0:10:30.40,Default,,0000,0000,0000,,One of the things I am going to\Ndo is I'm going to take the pie
Dialogue: 0,0:10:30.40,0:10:32.42,Default,,0000,0000,0000,,out from inside the integration
Dialogue: 0,0:10:32.42,0:10:36.82,Default,,0000,0000,0000,,sign. This is because the pie is\Na constant and I don't want it
Dialogue: 0,0:10:36.82,0:10:40.52,Default,,0000,0000,0000,,to get in the way of being able\Nto do this integral. Don't want
Dialogue: 0,0:10:40.52,0:10:43.97,Default,,0000,0000,0000,,it to be a distraction 'cause I\Nknow it's a constant. I know
Dialogue: 0,0:10:43.97,0:10:46.62,Default,,0000,0000,0000,,it's multiplying everything\Nthere, so I'm going to take it
Dialogue: 0,0:10:46.62,0:10:50.06,Default,,0000,0000,0000,,outside the integral so that I\Ndon't get in my way. It doesn't
Dialogue: 0,0:10:50.06,0:10:51.39,Default,,0000,0000,0000,,confuse what's going to happen.
Dialogue: 0,0:10:52.31,0:10:53.63,Default,,0000,0000,0000,,So that's π.
Dialogue: 0,0:10:54.26,0:10:58.75,Default,,0000,0000,0000,,And now let's have a look at the\Nintegration ask where it is just
Dialogue: 0,0:10:58.75,0:11:02.61,Default,,0000,0000,0000,,a constant. So when we integrate\Nit, we get our squared X.
Dialogue: 0,0:11:03.63,0:11:08.30,Default,,0000,0000,0000,,Minus, because we've got minus\Nthere. Now we integrate X
Dialogue: 0,0:11:08.30,0:11:14.84,Default,,0000,0000,0000,,squared, so that's add 1 to the\Nindex X cubed and divide by the
Dialogue: 0,0:11:14.84,0:11:21.09,Default,,0000,0000,0000,,new index. Now we need to\Nevaluate this integral between
Dialogue: 0,0:11:21.09,0:11:23.99,Default,,0000,0000,0000,,the two limits, minus R&R.
Dialogue: 0,0:11:25.18,0:11:31.13,Default,,0000,0000,0000,,Equals. The pie\Noutside now need to substitute
Dialogue: 0,0:11:31.13,0:11:35.50,Default,,0000,0000,0000,,these limits in intern. Let me\Nset up a big bracket.
Dialogue: 0,0:11:36.15,0:11:37.78,Default,,0000,0000,0000,,And will put in.
Dialogue: 0,0:11:38.53,0:11:44.96,Default,,0000,0000,0000,,All so I've asked squared times\Nby X and that gives me R cubed
Dialogue: 0,0:11:44.96,0:11:47.71,Default,,0000,0000,0000,,because X is equal to R.
Dialogue: 0,0:11:48.23,0:11:55.10,Default,,0000,0000,0000,,Minus now I've got X cubed, so\Nthat's going to be when X equals
Dialogue: 0,0:11:55.10,0:11:57.07,Default,,0000,0000,0000,,RR cubed over 3.
Dialogue: 0,0:11:57.72,0:12:02.59,Default,,0000,0000,0000,,Close the bracket minus and open\Nanother square bracket and this
Dialogue: 0,0:12:02.59,0:12:08.35,Default,,0000,0000,0000,,time we're putting minus are in,\Nso we R-squared times by X which
Dialogue: 0,0:12:08.35,0:12:12.78,Default,,0000,0000,0000,,is minus R and that gives us\Nminus R cubed.
Dialogue: 0,0:12:13.96,0:12:21.38,Default,,0000,0000,0000,,Minus and now I've got X cubed\Nover three, and this time X is
Dialogue: 0,0:12:21.38,0:12:26.15,Default,,0000,0000,0000,,minus are, so that's minus R\Ncubed over 3.
Dialogue: 0,0:12:26.15,0:12:29.76,Default,,0000,0000,0000,,Close the square bracket and\Nclose the big bracket.
Dialogue: 0,0:12:30.41,0:12:33.71,Default,,0000,0000,0000,,Equals. Pie.
Dialogue: 0,0:12:35.13,0:12:38.03,Default,,0000,0000,0000,,Big bracket. Now.
Dialogue: 0,0:12:38.55,0:12:43.64,Default,,0000,0000,0000,,Let's workout this one. We've R\Ncubed takeaway R cubed over
Dialogue: 0,0:12:43.64,0:12:49.66,Default,,0000,0000,0000,,three, so that's our cubed minus\N1/3 of our cubes leaves us with
Dialogue: 0,0:12:49.66,0:12:51.51,Default,,0000,0000,0000,,2/3 of our cubed.
Dialogue: 0,0:12:52.48,0:12:54.87,Default,,0000,0000,0000,,Minus that minus sign.
Dialogue: 0,0:12:55.65,0:13:01.04,Default,,0000,0000,0000,,Now let's have a look at this.\NWe've minus R, cubed minus, and
Dialogue: 0,0:13:01.04,0:13:03.95,Default,,0000,0000,0000,,then I've got R minus R cubed.
Dialogue: 0,0:13:04.76,0:13:09.31,Default,,0000,0000,0000,,Now a minus sign multiplied by\Nitself three times gives us
Dialogue: 0,0:13:09.31,0:13:15.11,Default,,0000,0000,0000,,minus with that minus gives us a\Nplus, so I have minus R cubed
Dialogue: 0,0:13:15.11,0:13:20.49,Default,,0000,0000,0000,,plus R cubed over three, which\Ngives me minus 2/3 of our cubed,
Dialogue: 0,0:13:20.49,0:13:25.87,Default,,0000,0000,0000,,so that bit there in the round\Nbrackets is exactly the same as
Dialogue: 0,0:13:25.87,0:13:30.84,Default,,0000,0000,0000,,that, but there in the square\Nbrackets and let's close off the
Dialogue: 0,0:13:30.84,0:13:36.64,Default,,0000,0000,0000,,curly brackets and now I have a\Nminus and minus gives me a plus.
Dialogue: 0,0:13:37.94,0:13:43.61,Default,,0000,0000,0000,,Just get the answer in here will\Nbe 2/3 are cubed, plus 2/3 are
Dialogue: 0,0:13:43.61,0:13:48.47,Default,,0000,0000,0000,,cubes. That'll be 4 thirds.\NLet's put the pie in our cube
Dialogue: 0,0:13:48.47,0:13:53.33,Default,,0000,0000,0000,,and that's the volume of the\Nsphere that we started off with,
Dialogue: 0,0:13:53.33,0:13:58.19,Default,,0000,0000,0000,,so we've been able to show by\Nusing this calculus that the
Dialogue: 0,0:13:58.19,0:14:01.02,Default,,0000,0000,0000,,volume of a sphere is 4 thirds
Dialogue: 0,0:14:01.02,0:14:06.31,Default,,0000,0000,0000,,Paillard cubed. The other\Nexample that we mentioned at the
Dialogue: 0,0:14:06.31,0:14:11.91,Default,,0000,0000,0000,,very beginning was a cone. So\Nagain, let's have a look at
Dialogue: 0,0:14:11.91,0:14:14.71,Default,,0000,0000,0000,,finding the volume of a cone.
Dialogue: 0,0:14:15.39,0:14:18.91,Default,,0000,0000,0000,,So we take a straight line that\Ngoes through the origin.
Dialogue: 0,0:14:21.25,0:14:26.81,Default,,0000,0000,0000,,Will identify the X values is\Ngoing from North to HH is the
Dialogue: 0,0:14:26.81,0:14:32.81,Default,,0000,0000,0000,,height of the code and will look\Nat the base radius of the cold
Dialogue: 0,0:14:32.81,0:14:38.37,Default,,0000,0000,0000,,which is our and then when we\Nrotate this line around the X
Dialogue: 0,0:14:38.37,0:14:42.22,Default,,0000,0000,0000,,axis through 360 degrees we will\Nget a cone.
Dialogue: 0,0:14:42.92,0:14:48.09,Default,,0000,0000,0000,,Now what's the equation of\Nthis line? Well, it's a
Dialogue: 0,0:14:48.09,0:14:53.26,Default,,0000,0000,0000,,straight line and it goes\Nthrough the origin, so it
Dialogue: 0,0:14:53.26,0:14:59.46,Default,,0000,0000,0000,,must be why equals MX, where\NM is the gradient. What is
Dialogue: 0,0:14:59.46,0:15:05.15,Default,,0000,0000,0000,,the gradient of the line? The\Ngradient of the line is
Dialogue: 0,0:15:05.15,0:15:11.36,Default,,0000,0000,0000,,defined to be the tangent of\Nthe angle that the line makes
Dialogue: 0,0:15:11.36,0:15:13.42,Default,,0000,0000,0000,,with the X axis.
Dialogue: 0,0:15:15.04,0:15:21.63,Default,,0000,0000,0000,,Theater so M equals 10 theater.\NBut we can see that we have a
Dialogue: 0,0:15:21.63,0:15:26.82,Default,,0000,0000,0000,,right angle triangle here. The\Nopposite side to the angle is
Dialogue: 0,0:15:26.82,0:15:33.41,Default,,0000,0000,0000,,our and the adjacent site to the\Nangle is H and so Tan Theater
Dialogue: 0,0:15:33.41,0:15:39.06,Default,,0000,0000,0000,,which is opposite over adjacent\Nis are over H. So this means
Dialogue: 0,0:15:39.06,0:15:45.66,Default,,0000,0000,0000,,that the equation of our line is\NY equals R over H times by.
Dialogue: 0,0:15:45.67,0:15:52.62,Default,,0000,0000,0000,,X. So now we've established\Nthe equation of the line. Now in
Dialogue: 0,0:15:52.62,0:15:59.34,Default,,0000,0000,0000,,a position to be able to rotate\Nit about the X axis. So let's
Dialogue: 0,0:15:59.34,0:16:05.10,Default,,0000,0000,0000,,again right down our formula for\Nthe volume of the cone that
Dialogue: 0,0:16:05.10,0:16:11.82,Default,,0000,0000,0000,,we're going to get. V equals the\Nintegral from A to B. Those are
Dialogue: 0,0:16:11.82,0:16:14.70,Default,,0000,0000,0000,,RX values. Remember of Pi Y
Dialogue: 0,0:16:14.70,0:16:20.99,Default,,0000,0000,0000,,squared DX. And let's now\Nidentify all of these pieces. A
Dialogue: 0,0:16:20.99,0:16:27.71,Default,,0000,0000,0000,,is the lower value of X, and\Nthat's at zero the origin. So X
Dialogue: 0,0:16:27.71,0:16:32.99,Default,,0000,0000,0000,,IS0B is the higher value of X,\Nso that's H there.
Dialogue: 0,0:16:33.74,0:16:39.66,Default,,0000,0000,0000,,Pie. And now we want Y\Nsquared and we established that
Dialogue: 0,0:16:39.66,0:16:46.54,Default,,0000,0000,0000,,why was equal to R over H times\Nby X, so that's our over H times
Dialogue: 0,0:16:46.54,0:16:52.56,Default,,0000,0000,0000,,by X and we have to square it.\NThat is to be integrated with
Dialogue: 0,0:16:52.56,0:16:58.15,Default,,0000,0000,0000,,respect to X. Do remember to\NSquare the Y. It's a very common
Dialogue: 0,0:16:58.15,0:17:03.31,Default,,0000,0000,0000,,error just to put in Y and then\Nintegrate it without squaring
Dialogue: 0,0:17:03.31,0:17:05.89,Default,,0000,0000,0000,,first. OK, turn over the page.
Dialogue: 0,0:17:05.93,0:17:10.54,Default,,0000,0000,0000,,And as we do so, we're going to\Nwrite down what we've got here
Dialogue: 0,0:17:10.54,0:17:15.90,Default,,0000,0000,0000,,to integrate. So our volume is\Nequal to, and again, I'm going
Dialogue: 0,0:17:15.90,0:17:20.89,Default,,0000,0000,0000,,to take the pie outside the\Nintegration sign so that I don't
Dialogue: 0,0:17:20.89,0:17:25.88,Default,,0000,0000,0000,,get confused by it doesn't get\Nin the way of the integration
Dialogue: 0,0:17:25.88,0:17:32.54,Default,,0000,0000,0000,,limits are from North to H and I\Nneed to square the Y. If we just
Dialogue: 0,0:17:32.54,0:17:38.78,Default,,0000,0000,0000,,look back, will see that the why\Nwas RX over age. So if I square
Dialogue: 0,0:17:38.78,0:17:41.69,Default,,0000,0000,0000,,it, it will be our squared X
Dialogue: 0,0:17:41.69,0:17:44.68,Default,,0000,0000,0000,,squared. Over H squared.
Dialogue: 0,0:17:45.25,0:17:52.55,Default,,0000,0000,0000,,So let's do that all squared\NX squared over 8 squared to
Dialogue: 0,0:17:52.55,0:17:56.19,Default,,0000,0000,0000,,be integrated with respect to X.
Dialogue: 0,0:17:56.79,0:17:59.67,Default,,0000,0000,0000,,So π.
Dialogue: 0,0:18:00.51,0:18:06.69,Default,,0000,0000,0000,,Square bracket.\NAll squared over 8 squared.
Dialogue: 0,0:18:06.69,0:18:08.75,Default,,0000,0000,0000,,That's just a constant.
Dialogue: 0,0:18:10.13,0:18:16.88,Default,,0000,0000,0000,,And X squared we integrate the X\Nsquared and one to the index
Dialogue: 0,0:18:16.88,0:18:23.62,Default,,0000,0000,0000,,that's X cubed and divide by the\Nnew index. So we divide by
Dialogue: 0,0:18:23.62,0:18:29.85,Default,,0000,0000,0000,,three. This is to be evaluated\Nbetween North and H equals Π.
Dialogue: 0,0:18:30.59,0:18:37.27,Default,,0000,0000,0000,,The H in first, so\Nwe R-squared over H squared.
Dialogue: 0,0:18:37.88,0:18:41.50,Default,,0000,0000,0000,,Times by H cubed over
Dialogue: 0,0:18:41.50,0:18:48.30,Default,,0000,0000,0000,,3. So there's our\Nfirst bracket, minus the second
Dialogue: 0,0:18:48.30,0:18:55.49,Default,,0000,0000,0000,,one. When we put zero in\NR-squared over H squared times
Dialogue: 0,0:18:55.49,0:18:58.10,Default,,0000,0000,0000,,by zero over 3.
Dialogue: 0,0:18:58.11,0:19:01.33,Default,,0000,0000,0000,,And of course, that means this\Nend bracket is 0.
Dialogue: 0,0:19:01.87,0:19:07.12,Default,,0000,0000,0000,,So we only need look at this. We\Ncan see that we have an H
Dialogue: 0,0:19:07.12,0:19:10.97,Default,,0000,0000,0000,,squared here which will cancel\Nwith the H Cube there, leaving
Dialogue: 0,0:19:10.97,0:19:13.77,Default,,0000,0000,0000,,us with H and so will have 1/3.
Dialogue: 0,0:19:14.29,0:19:20.58,Default,,0000,0000,0000,,Π R squared H and that's\Nthe standard formula for the
Dialogue: 0,0:19:20.58,0:19:22.87,Default,,0000,0000,0000,,volume of a cone.
Dialogue: 0,0:19:23.94,0:19:29.30,Default,,0000,0000,0000,,OK, we've had a look at two\Nfairly standard formula that we
Dialogue: 0,0:19:29.30,0:19:34.22,Default,,0000,0000,0000,,knew already and we've seen how\Nwe can calculate them using
Dialogue: 0,0:19:34.22,0:19:38.69,Default,,0000,0000,0000,,calculus. But of course these\Naren't general curves that were
Dialogue: 0,0:19:38.69,0:19:44.06,Default,,0000,0000,0000,,fairly specific cases, so this\Ntime let's have a look at a
Dialogue: 0,0:19:44.06,0:19:45.40,Default,,0000,0000,0000,,slightly different question.
Dialogue: 0,0:19:46.10,0:19:53.40,Default,,0000,0000,0000,,This is a question that occurs\Nin many textbooks or over style
Dialogue: 0,0:19:53.40,0:19:56.44,Default,,0000,0000,0000,,that occurs in many textbooks.
Dialogue: 0,0:19:57.03,0:20:04.45,Default,,0000,0000,0000,,Take. Of\NY equals X squared minus one.
Dialogue: 0,0:20:08.47,0:20:10.74,Default,,0000,0000,0000,,And find the volume.
Dialogue: 0,0:20:14.04,0:20:17.51,Default,,0000,0000,0000,,Of the solid.
Dialogue: 0,0:20:18.70,0:20:24.33,Default,,0000,0000,0000,,Revolution.
Dialogue: 0,0:20:24.54,0:20:30.76,Default,,0000,0000,0000,,When?
Dialogue: 0,0:20:31.35,0:20:33.24,Default,,0000,0000,0000,,The area.
Dialogue: 0,0:20:33.75,0:20:40.99,Default,,0000,0000,0000,,Contained.\NBy
Dialogue: 0,0:20:40.99,0:20:43.36,Default,,0000,0000,0000,,the
Dialogue: 0,0:20:43.36,0:20:45.72,Default,,0000,0000,0000,,curve.
Dialogue: 0,0:20:46.40,0:20:52.51,Default,,0000,0000,0000,,And.\NThe X axis.
Dialogue: 0,0:20:53.02,0:21:00.69,Default,,0000,0000,0000,,Is rotated about\Nthe X axis.
Dialogue: 0,0:21:02.34,0:21:10.12,Default,,0000,0000,0000,,Through\N360
Dialogue: 0,0:21:10.12,0:21:12.19,Default,,0000,0000,0000,,degrees.
Dialogue: 0,0:21:13.28,0:21:17.97,Default,,0000,0000,0000,,So here we've got our curve and\Nwe want the volume of solid of
Dialogue: 0,0:21:17.97,0:21:21.66,Default,,0000,0000,0000,,revolution when the area\Ncontained by the curve on the X
Dialogue: 0,0:21:21.66,0:21:27.08,Default,,0000,0000,0000,,axis. That suggests that we\Nreally need to look at a picture
Dialogue: 0,0:21:27.08,0:21:32.74,Default,,0000,0000,0000,,of this curve. We really need to\Nsort out where it is in respect
Dialogue: 0,0:21:32.74,0:21:38.42,Default,,0000,0000,0000,,of the X axis, so let's have a\Nlook. We've got Y equals X
Dialogue: 0,0:21:38.42,0:21:40.44,Default,,0000,0000,0000,,squared minus one we want.
Dialogue: 0,0:21:41.16,0:21:46.75,Default,,0000,0000,0000,,Have a little sketch of this\Ncurve. We want to know what it's
Dialogue: 0,0:21:46.75,0:21:49.33,Default,,0000,0000,0000,,doing. Y equals X squared minus
Dialogue: 0,0:21:49.33,0:21:54.68,Default,,0000,0000,0000,,one. But one of the things that\Nwill be useful to know is where
Dialogue: 0,0:21:54.68,0:21:59.10,Default,,0000,0000,0000,,does it actually cross the X\Naxis? So in order to do that on
Dialogue: 0,0:21:59.10,0:22:01.30,Default,,0000,0000,0000,,the X axis, we know that Y
Dialogue: 0,0:22:01.30,0:22:07.73,Default,,0000,0000,0000,,equals 0. So let's write that\Ndown. Why is 0 so X squared
Dialogue: 0,0:22:07.73,0:22:13.84,Default,,0000,0000,0000,,minus one more speech 0 where it\Ncrosses the X axis so we can
Dialogue: 0,0:22:13.84,0:22:19.09,Default,,0000,0000,0000,,factorize this and this is X\Nsquared minus one. So it's the
Dialogue: 0,0:22:19.09,0:22:23.90,Default,,0000,0000,0000,,difference of two squares, so\Nwill factorize as X minus one
Dialogue: 0,0:22:23.90,0:22:25.64,Default,,0000,0000,0000,,and X plus one.
Dialogue: 0,0:22:26.26,0:22:32.48,Default,,0000,0000,0000,,Equals 0. This is\Na quadratic equation that we
Dialogue: 0,0:22:32.48,0:22:37.26,Default,,0000,0000,0000,,factorized into two separate\Nfactors. So if the two factors
Dialogue: 0,0:22:37.26,0:22:42.99,Default,,0000,0000,0000,,multiplied together give zero,\Nthen one of them is 0 or the
Dialogue: 0,0:22:42.99,0:22:48.73,Default,,0000,0000,0000,,other one is 0, or they're\Nboth zero, and this tells us
Dialogue: 0,0:22:48.73,0:22:54.94,Default,,0000,0000,0000,,that then X equals 1 or X\Nequals minus one. So we know
Dialogue: 0,0:22:54.94,0:22:59.24,Default,,0000,0000,0000,,that the curve goes through\Nthere and through there.
Dialogue: 0,0:23:00.69,0:23:06.88,Default,,0000,0000,0000,,What else do we know about this?\NWell, what if we set X is equal
Dialogue: 0,0:23:06.88,0:23:11.84,Default,,0000,0000,0000,,to 0? What if we investigate\Nwhat happens when X is zero?
Dialogue: 0,0:23:11.84,0:23:17.62,Default,,0000,0000,0000,,Will straight away we can see\Nwhen X is zero, Y is equal to
Dialogue: 0,0:23:17.62,0:23:20.10,Default,,0000,0000,0000,,minus one, so the curves down
Dialogue: 0,0:23:20.10,0:23:25.45,Default,,0000,0000,0000,,there. So we can tell now that\Nthe curve must look something.
Dialogue: 0,0:23:27.12,0:23:33.13,Default,,0000,0000,0000,,Like That And this is\Nthe area that is trapped between
Dialogue: 0,0:23:33.13,0:23:38.60,Default,,0000,0000,0000,,the X axis and the curve. So\Nthis is the area that he's going
Dialogue: 0,0:23:38.60,0:23:43.30,Default,,0000,0000,0000,,to be rotated through 360\Ndegrees and that we have to find
Dialogue: 0,0:23:43.30,0:23:47.99,Default,,0000,0000,0000,,the volume of that solid of\NRevolution. So now we've got a
Dialogue: 0,0:23:47.99,0:23:52.68,Default,,0000,0000,0000,,picture and we know what we're\Ndoing. We're in a position to
Dialogue: 0,0:23:52.68,0:23:53.85,Default,,0000,0000,0000,,start the calculation.
Dialogue: 0,0:23:55.05,0:24:02.55,Default,,0000,0000,0000,,So we begin with V\Nequals the integral from A
Dialogue: 0,0:24:02.55,0:24:06.30,Default,,0000,0000,0000,,to B of Pi Y
Dialogue: 0,0:24:06.30,0:24:09.59,Default,,0000,0000,0000,,squared DX. Equals.
Dialogue: 0,0:24:10.55,0:24:15.52,Default,,0000,0000,0000,,First of all, let's identify the\Nlimits of integration.
Dialogue: 0,0:24:15.82,0:24:17.54,Default,,0000,0000,0000,,This is where the curve.
Dialogue: 0,0:24:18.55,0:24:25.05,Default,,0000,0000,0000,,Passed through the X axis so the\Nlimits of integration are minus
Dialogue: 0,0:24:25.05,0:24:32.64,Default,,0000,0000,0000,,one and one the values of X\Nwhere the curve cuts the X axis
Dialogue: 0,0:24:32.64,0:24:39.15,Default,,0000,0000,0000,,pie. And Y squared? Well, let's\Njust be careful here. We know
Dialogue: 0,0:24:39.15,0:24:42.10,Default,,0000,0000,0000,,that. Why is X squared minus
Dialogue: 0,0:24:42.10,0:24:45.62,Default,,0000,0000,0000,,one? So we can write that in.
Dialogue: 0,0:24:47.63,0:24:53.18,Default,,0000,0000,0000,,X squared minus one and we\Nremember to square it
Dialogue: 0,0:24:53.18,0:24:59.28,Default,,0000,0000,0000,,integrated with respect to X.\NWe got two square out this
Dialogue: 0,0:24:59.28,0:25:05.39,Default,,0000,0000,0000,,bracket. We got to multiply\Ntown so integral of Pi. Notice
Dialogue: 0,0:25:05.39,0:25:10.38,Default,,0000,0000,0000,,I'm moving the pie outside the\Nintegration sign again.
Dialogue: 0,0:25:11.60,0:25:18.32,Default,,0000,0000,0000,,And I want this squared, so let\Nme do it here in full so we can
Dialogue: 0,0:25:18.32,0:25:22.94,Default,,0000,0000,0000,,see where all the terms come\Nfrom. That's what the square
Dialogue: 0,0:25:22.94,0:25:25.04,Default,,0000,0000,0000,,means. Multiply the bracket by
Dialogue: 0,0:25:25.04,0:25:31.22,Default,,0000,0000,0000,,itself. So with X squared times\Nby X squared, that gives us X to
Dialogue: 0,0:25:31.22,0:25:32.39,Default,,0000,0000,0000,,the power 4.
Dialogue: 0,0:25:32.93,0:25:39.13,Default,,0000,0000,0000,,We've X squared times by minus\None gives us minus X squared.
Dialogue: 0,0:25:39.79,0:25:45.22,Default,,0000,0000,0000,,We've minus one times by X\Nsquared. Gives us minus X
Dialogue: 0,0:25:45.22,0:25:51.31,Default,,0000,0000,0000,,squared. And we've minus one\Ntimes by minus one gives us plus
Dialogue: 0,0:25:51.31,0:25:57.19,Default,,0000,0000,0000,,one and this middle term we have\Nminus X squared minus X squared.
Dialogue: 0,0:25:57.19,0:26:04.42,Default,,0000,0000,0000,,So we've X to the 4th minus two\NX squared plus one. So this is Y
Dialogue: 0,0:26:04.42,0:26:10.75,Default,,0000,0000,0000,,squared, which we can write in\Nhere X to the 4th, minus two X
Dialogue: 0,0:26:10.75,0:26:12.56,Default,,0000,0000,0000,,squared plus one DX.
Dialogue: 0,0:26:13.42,0:26:16.95,Default,,0000,0000,0000,,Notice that is important to make\Nsure you get this square right,
Dialogue: 0,0:26:16.95,0:26:21.65,Default,,0000,0000,0000,,and so it's a good idea to do it\Nat one site to writing out in
Dialogue: 0,0:26:21.65,0:26:23.71,Default,,0000,0000,0000,,full, just to make sure you get
Dialogue: 0,0:26:23.71,0:26:30.84,Default,,0000,0000,0000,,it right. OK, I'll turn over the\Npage and will go through this
Dialogue: 0,0:26:30.84,0:26:33.45,Default,,0000,0000,0000,,integration. So our volume V.
Dialogue: 0,0:26:34.20,0:26:41.18,Default,,0000,0000,0000,,Equals π times the integral\Nfrom minus one to One
Dialogue: 0,0:26:41.18,0:26:48.16,Default,,0000,0000,0000,,X to the 4th, minus\Ntwo X squared plus one
Dialogue: 0,0:26:48.16,0:26:51.65,Default,,0000,0000,0000,,with respect to X equals
Dialogue: 0,0:26:51.65,0:26:58.50,Default,,0000,0000,0000,,π. Now we'll do the integration.\NThe integral of X to the 4th is
Dialogue: 0,0:26:58.50,0:27:03.70,Default,,0000,0000,0000,,X to the fifth. Adding one to\Nthe index and dividing by the
Dialogue: 0,0:27:03.70,0:27:08.90,Default,,0000,0000,0000,,new index. So we have X to the\Nfifth over 5 - 2.
Dialogue: 0,0:27:09.68,0:27:14.49,Default,,0000,0000,0000,,X squared when we integrate it\Nis X cubed over three. Adding
Dialogue: 0,0:27:14.49,0:27:19.70,Default,,0000,0000,0000,,one to the index and dividing by\Nthe new index plus and the
Dialogue: 0,0:27:19.70,0:27:24.92,Default,,0000,0000,0000,,integral of one is just X and\Nthis is between minus one and
Dialogue: 0,0:27:24.92,0:27:32.48,Default,,0000,0000,0000,,one. Substituting the limits one\Nand we put it in first.
Dialogue: 0,0:27:32.48,0:27:35.72,Default,,0000,0000,0000,,That gives us a fifth.
Dialogue: 0,0:27:36.64,0:27:43.69,Default,,0000,0000,0000,,Minus 2/3 because all of these\Npowers of X or just one plus
Dialogue: 0,0:27:43.69,0:27:50.80,Default,,0000,0000,0000,,one. Minus. Now\Nwe have to be a bit more
Dialogue: 0,0:27:50.80,0:27:55.59,Default,,0000,0000,0000,,careful this time, 'cause\Nwe've got minus one. So when
Dialogue: 0,0:27:55.59,0:28:02.30,Default,,0000,0000,0000,,we put minus one in minus one\Nto the 5th is minus one over
Dialogue: 0,0:28:02.30,0:28:09.00,Default,,0000,0000,0000,,5 - 1/5 - 2/3 of. Now X\Nis cubed, so it's minus one
Dialogue: 0,0:28:09.00,0:28:14.27,Default,,0000,0000,0000,,cubed and that is minus one\Nand then plus. And we're
Dialogue: 0,0:28:14.27,0:28:17.63,Default,,0000,0000,0000,,putting minus one in minus\None there.
Dialogue: 0,0:28:19.73,0:28:26.63,Default,,0000,0000,0000,,Equals π? I just need to squeeze\Nin a bracket just to close that
Dialogue: 0,0:28:26.63,0:28:31.07,Default,,0000,0000,0000,,bracket. Now let's workout each\Nof these two brackets.
Dialogue: 0,0:28:32.32,0:28:37.84,Default,,0000,0000,0000,,And we need some knowledge of\Nfractions to be able to do this,
Dialogue: 0,0:28:37.84,0:28:42.94,Default,,0000,0000,0000,,and by the looks of it, we're\Ngoing to need a common
Dialogue: 0,0:28:42.94,0:28:48.04,Default,,0000,0000,0000,,denominator, so let's make them\Nall over 1515, because 15 is 5
Dialogue: 0,0:28:48.04,0:28:53.14,Default,,0000,0000,0000,,times by three and both five and\Nthree will divide exactly into
Dialogue: 0,0:28:53.14,0:28:53.57,Default,,0000,0000,0000,,15.
Dialogue: 0,0:28:54.25,0:29:03.04,Default,,0000,0000,0000,,So.\N5 into 15 goes three, so 1/5
Dialogue: 0,0:29:03.04,0:29:06.24,Default,,0000,0000,0000,,is the same as three fifteenths.
Dialogue: 0,0:29:06.94,0:29:13.78,Default,,0000,0000,0000,,3 into 15 goes 5 and so five\Ntimes by two is ten. 10:15 set
Dialogue: 0,0:29:13.78,0:29:20.62,Default,,0000,0000,0000,,the same as 2/3 and one is a\Nwhole one, so it must be 15
Dialogue: 0,0:29:20.62,0:29:22.85,Default,,0000,0000,0000,,fifteenths. Minus.
Dialogue: 0,0:29:23.95,0:29:30.02,Default,,0000,0000,0000,,No.\NGoing to convert these into
Dialogue: 0,0:29:30.02,0:29:34.54,Default,,0000,0000,0000,,Fifteenths as well, but we need\Nto be a bit careful about the
Dialogue: 0,0:29:34.54,0:29:36.28,Default,,0000,0000,0000,,signs while we're doing it.
Dialogue: 0,0:29:36.29,0:29:41.41,Default,,0000,0000,0000,,So this is minus 1/5. So\Nin terms of 15, so it must
Dialogue: 0,0:29:41.41,0:29:42.99,Default,,0000,0000,0000,,be minus three fifteens.
Dialogue: 0,0:29:44.04,0:29:50.34,Default,,0000,0000,0000,,Now I have a minus and minus\Nthere that will make that a plus
Dialogue: 0,0:29:50.34,0:29:53.49,Default,,0000,0000,0000,,and so it's going to be plus
Dialogue: 0,0:29:53.49,0:29:59.97,Default,,0000,0000,0000,,1050. And here I've got plus and\Nminus, so that's going to be a
Dialogue: 0,0:29:59.97,0:30:02.74,Default,,0000,0000,0000,,minus and it will be 1550.
Dialogue: 0,0:30:03.15,0:30:05.88,Default,,0000,0000,0000,,Equals
Dialogue: 0,0:30:05.88,0:30:13.20,Default,,0000,0000,0000,,π. Times\NNow let's have a look at
Dialogue: 0,0:30:13.20,0:30:18.36,Default,,0000,0000,0000,,these. We three fifteenths\Ntakeaway 10 fifteenths plus
Dialogue: 0,0:30:18.36,0:30:25.36,Default,,0000,0000,0000,,1515's. So the three and\Nthe 15 give us 18 takeaway
Dialogue: 0,0:30:25.36,0:30:27.66,Default,,0000,0000,0000,,10. That's eight fifteenths.
Dialogue: 0,0:30:27.93,0:30:31.46,Default,,0000,0000,0000,,So that's got that one sorted
Dialogue: 0,0:30:31.46,0:30:37.61,Default,,0000,0000,0000,,minus. Now let's have a look\Nat this. We've minus three
Dialogue: 0,0:30:37.61,0:30:42.99,Default,,0000,0000,0000,,fifteenths minus 1515, so that's\Nminus 18 fifteenths, plus 10 of
Dialogue: 0,0:30:42.99,0:30:48.86,Default,,0000,0000,0000,,them. That gives us minus eight\N15th, so minus 8 Fifteenths. And
Dialogue: 0,0:30:48.86,0:30:53.26,Default,,0000,0000,0000,,we can just finish this\Ncalculation off now, because
Dialogue: 0,0:30:53.26,0:30:58.15,Default,,0000,0000,0000,,we've 8 fifteenths minus minus\N8, fifteenths gives us 16
Dialogue: 0,0:30:58.15,0:31:03.04,Default,,0000,0000,0000,,Fifteenths and π times by π. So\Nthere's our answer.
Dialogue: 0,0:31:03.09,0:31:05.04,Default,,0000,0000,0000,,And that's our calculation.
Dialogue: 0,0:31:05.73,0:31:13.01,Default,,0000,0000,0000,,There is one more thing that we\Njust need to have a quick look
Dialogue: 0,0:31:13.01,0:31:18.73,Default,,0000,0000,0000,,at. If we can rotate a curve\Nabout the X axis.
Dialogue: 0,0:31:20.46,0:31:23.12,Default,,0000,0000,0000,,Spinning it round here through
Dialogue: 0,0:31:23.12,0:31:29.02,Default,,0000,0000,0000,,360 degrees. It's actually no\Nreason why we shouldn't rotate a
Dialogue: 0,0:31:29.02,0:31:35.12,Default,,0000,0000,0000,,curve about the Y axis. Spinning\Nit around the Y axis. So let's
Dialogue: 0,0:31:35.12,0:31:36.99,Default,,0000,0000,0000,,say we did that.
Dialogue: 0,0:31:37.57,0:31:42.06,Default,,0000,0000,0000,,Between the values of Y which\Nare Y equals C.
Dialogue: 0,0:31:42.56,0:31:45.07,Default,,0000,0000,0000,,And why equals day?
Dialogue: 0,0:31:46.08,0:31:49.03,Default,,0000,0000,0000,,Well, there really is no\Ndifference between spinning it
Dialogue: 0,0:31:49.03,0:31:53.30,Default,,0000,0000,0000,,around the Y axis and spinning\Nit around the X axis. The way
Dialogue: 0,0:31:53.30,0:31:57.23,Default,,0000,0000,0000,,that we calculate it is going to\Nbe the same, except because
Dialogue: 0,0:31:57.23,0:32:01.17,Default,,0000,0000,0000,,we've spun it around the Y axis\Ninstead of the X axis.
Dialogue: 0,0:32:01.94,0:32:08.13,Default,,0000,0000,0000,,The X's and wiser going to\Ninterchange, and so we end up
Dialogue: 0,0:32:08.13,0:32:15.36,Default,,0000,0000,0000,,here to the idea that the volume\Nin this case is from Y equals
Dialogue: 0,0:32:15.36,0:32:22.58,Default,,0000,0000,0000,,C to Y equals D of Pi\NX squared DY, and in the same
Dialogue: 0,0:32:22.58,0:32:28.26,Default,,0000,0000,0000,,way as we had expressions for Y\Nin terms of X.
Dialogue: 0,0:32:29.06,0:32:34.51,Default,,0000,0000,0000,,In order to do this, we would\Nhave to have expressions for X
Dialogue: 0,0:32:34.51,0:32:39.95,Default,,0000,0000,0000,,in terms of Y and then we can\Nintegrate with respect to Y,
Dialogue: 0,0:32:39.95,0:32:43.72,Default,,0000,0000,0000,,substituting the values of Y and\Nobtain the volumes.
Dialogue: 0,0:32:44.29,0:32:48.63,Default,,0000,0000,0000,,So that's volumes of\Nsolids of Revolution.