00:00Hi guys, so this is
00:02my third and final video
00:03and volume of revolution in
00:05this lesson What we're gonna
00:07look at is the volume
00:09of revolution between two curves
00:11now what that means is
00:12a bit of these What
00:13if you remember the area
00:14between two curves? So here
00:15this was the area between
00:16two curves now imagine we
00:18revolve this around the x
00:20axis or the y axis,
00:21but let's look at the
00:21x axis first what you
00:24get is Well, you get
00:27some shape like
00:28this. So this is actually
00:29a torus and if this
00:30green if this green shape
00:32was a perfect circle you
00:33get this torus shape which
00:34is basically like a donut.
00:37So imagine if the x
00:39axis is going straight through
00:40the middle and yeah you
00:43get a hole in the
00:44middle of the shape like
00:45the hole in a donut.
00:46Similarly if you spun it
00:47around the y axis it
00:49would be a different type
00:50of shape but you have
00:51to kind of visualize what
00:53you would get. Now the
00:55formula
00:56are, well, they are not
00:59given in the formula booklet,
01:00but they're essentially just the
01:02volume underneath the purple curve
01:06minus the volume underneath the
01:09blue curve. And when I
01:10say volume in the volume
01:11of the of the rotation,
01:13so literally just this one
01:15minus this one, where f
01:16of x is the kind
01:16of the upper curve and
01:18g of x minus the
01:19lower curve. Note, look, well,
01:22you can take the pie
01:22out, but note it's
01:24f of x squared minus
01:26g of x squared, the
01:28common mistake is to do
01:31f of x minus g
01:33of x squared. I've seen
01:36that done many a time
01:38and this is obviously wrong
01:39because this is not the
01:40same as this. So just
01:43be very, very careful. This
01:44is when you, this is
01:45when you revolve around the
01:47x axis and this is
01:48the one where you revolve
01:49around the y axis. Now
01:51the limit
01:52That's the A and B
01:53a bit like the area
01:54between two curve limits. The
01:55limits are the intersection points.
01:59So for the, for the,
02:03when we're revolving around the
02:04x -axis, A and B
02:06are x values, so this
02:08would be A and this
02:10would be B. And when
02:11we're rotating or revolving around
02:13the y -axis, then those
02:15are y values. So that
02:17would be A and
02:20That would be B. Okay,
02:25so look, this isn't hugely
02:27difficult if you know how
02:28to do the normal volume
02:31of revolution. You're just subtracting
02:34them, but I am going
02:35to do a fairly tricky
02:38pass paper question that does
02:39just that. So it says
02:42the function f is defined
02:44by this for x agree
02:45the one of the function
02:46g is defined by this,
02:47the region or
02:48is bounded by the curves
02:50y equals f of x,
02:52g of x, and the
02:53lines y equals zero, x
02:54equals zero, and y equals
02:56nine. So firstly, let's just
02:58figure out what's actually going
03:00on. So which is which?
03:03So this one, hopefully you
03:06can see this is going
03:08to be x squared plus
03:08one, because that's your quadratic
03:10crossing at the y axis
03:12at one. So this is
03:14y equals x squared plus
03:16one and this is y
03:19equals x minus one squared.
03:24Then it says, and the
03:26region or is bound by
03:27y equals zero, that's the
03:29x -axis, x equals zero,
03:30that's the y -axis, and
03:31the line y equals nine.
03:34So it's this line here.
03:38Okay, so this region here
03:41in between the two curves
03:43and bounded
03:44by 0 and 9, this
03:46is my region or this
03:49is or and my limits
03:53are going to be 0
03:54and 9. Now we are
03:57rotating the region through 360
04:00degrees about the y axis
04:01that's going about the y
04:03axis. So a bit like,
04:05if we go back to
04:05this formula here, now we
04:10have we have to get
04:10x in terms of y
04:12but when we get
04:12like when we're getting the
04:14area between the curve and
04:17the y -axis. So we
04:18need to rearrange these functions
04:20to get x in terms
04:22of y. So first let's
04:25do y equals f of
04:27x first. So for y
04:29equals f of x, I'll
04:31just do it like this.
04:33So I'm gonna have y
04:34equals x minus one squared.
04:37Rearrange this. So I'm gonna
04:39have the square root of
04:40y squared both sides equals
04:42x minus 1 and then
04:44x equals the square root
04:46of y plus 1. And
04:48then for y equals g
04:52of x for y equals
04:56g of x, I'm going
04:56to have y equals x
05:02squared plus 1. x squared
05:05equals y minus 1 and
05:07x is equal
05:08to the square root of
05:10y minus 1. Okay, so
05:13now I can get the
05:14integral. So the volume of
05:16this thing rotated. So imagine
05:18now we're going to rotate
05:19this around. What he's looking
05:20for is, it's not like
05:23in my example where I
05:25did the volume around the
05:26y axis. I got the
05:27volume of the glass. So
05:29if we rotate this and
05:30we get a, we get
05:32a vase, he doesn't want
05:33the volume of the vase,
05:34he wants the volume of
05:35the clay used
05:36use a thing. I think
05:38the clay used to make
05:39the vase will be thick
05:40and it'll be this, this
05:41is kind of like the
05:41thickness of the vase. So
05:43it's like how much, how
05:44much clay will be used?
05:45That's where we're subtracting this,
05:48that makes sense. So the
05:50volume is pi times the
05:53integral from zero to nine.
05:55Those are my limits because
05:56it goes from zero, y
05:58equals zero to y equals
05:59nine. And then I have
06:02this squared,
06:04minus this square. So like
06:06f of x is the
06:07outer function. So it's going
06:08to be this one, minus
06:10this one. This is the
06:10outer function. This is the
06:12inner function. So I need
06:13to do root y plus
06:17one squared. And I need
06:20to do minus the square
06:22root of y minus one
06:27squared dy. Now this is
06:32a kind
06:33Acholated paper. Obviously, if it
06:36was non -calcative, you have
06:37to actually do the integral.
06:39But if it's a calculated
06:40paper, that's all you need
06:42to do. You need to
06:42just get this and then
06:44get your calculator and write
06:45down the answer. Now the
06:46only thing I'll say is
06:47your calculator will probably have
06:50difficulty dealing with this integrating
06:53this square root because there's
06:55a y minus 1. So
06:56I would just square this
06:57to begin with. So it'll
06:58be pi times
07:01The integral of, I can
07:04keep this square root of
07:05y plus 1 squared minus,
07:09you're squaring the square root,
07:10so it's just going to
07:11be y minus 1 dy
07:15like this. Okay, I'm going
07:18to put that into the
07:20calculator and I should just
07:22simply get my answer. So
07:24it's menu, I need a
07:26calculator, menu calculator,
07:29this numerical integral. So it
07:33goes from zero to nine.
07:34I don't forget to multiply
07:35by pi. I'm going to
07:36multiply by my answer by
07:37pi at the end. So
07:38zero to nine. I'll do
07:41the square root, the square.
07:44I can do it in
07:45y. I can do it
07:46all in y. The square
07:47root of y, actually I
07:49need another bracket. Square root
07:51of y plus one. Now,
07:54bear in mind, there's literally
07:54only one. I think this
07:56is a six mark question.
07:57And for the end for
07:58this next line it's only
07:59one mark. So this all
08:02just this here is actually
08:03worth five marks. So it's
08:05this squared minus y minus
08:09one, not squared because I've
08:12already squared it. Close the
08:13brackets, dy press enter and
08:18it's actually 54. And then
08:21I'm just going to multiply
08:22this by pi multiply by
08:24that's like
08:25It's probably like 54 .00
08:27or something. I doubt it's
08:29exactly 54, but it's not
08:30because it does one point.
08:31So answer times pi equals
08:38169 .6469 .6, 469 .6,
08:44469 .6, does it say
08:47meters or anything? No, it
08:49doesn't. So I'll just say
08:50units, units cubed.
08:53I could, if you wanted,
08:57write this as 170 units
09:00cubed to the nearest to
09:03three significant figures. But remember,
09:07always put down at least
09:10six and then you can
09:11put this down if you
09:13want. But if you doesn't
09:14actually see anything in the
09:14question, then you're fine. Okay,
09:17that's it. That's the volume
09:19of revolution between two curves.
09:22two formulae, not in the
09:26formulae booklet and you're essentially
09:27just subtracting your kind of
09:30original formulae that you are
09:31given in the formulae booklet.
09:33Hopefully that makes sense. Hope
09:35you enjoyed the lesson and
09:36see you in the next
09:37one.