How to Find Volumes of Revolution With Integration

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  • 0:06 Understanding…
  • 1:40 The Slicing Method
  • 3:43 The Disk Method
  • 4:52 The Washer Method
  • 7:33 Lesson Summary
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Lesson Transcript
Instructor: Kelly Sjol
Some shapes look the same as you rotate them, like the body of a football. In this lesson, learn how to find the volumes of shapes that have symmetry around an axis using the volume of revolution integration technique.

Understanding Generated Regions

A football can be used to explain the concept of generating regions
generated regions

Have you ever watched a really good quarterback throw a football, how it rotates along some axis? If you slice the football in half somewhere along that axis and take just the top half, as it rotates, this slice of the football will look like an entire football. If it rotates really fast, this region will 'create' the entire football. In math, we call this a generating region around an axis of rotation. This means, if I give you this generating region and axis of rotation and you watch this ball spin, you'll 'generate' an entire football. If you're the receiver looking at the ball coming at you, you see the region that's rotating around the axis as an entire football.

So let's say for a football, the generating region is given by the x-axis on the bottom and the function f(x)=2-(1/2)(x-2)^2 on the top. Say you want to find the volume of the football that has been generated by this rotating generating region. One way you can find the volume of a football is to slice it up.

The Slicing Method

Imagine a Nerf football that you just start slicing straight down from the back to the front. You slice it into many different intervals, turning the football into a whole bunch of disks. If you want to estimate the volume of the football, all you need to do is estimate the volume of each disk and add up all the disks, right?

The volume of one disk is equal to height times the cross-sectional area
Football Disk Volume

Let's estimate the volume of one disk. One disk has a volume of height times the cross-sectional area. The height is the thickness of the disk, or, of the slice you've taken of the Nerf football (which is delta*x), and the cross-sectional area is pi times the radius squared. In this particular case, our radius is given by the top part of our generating function, 2-(1/2)(x-2)^2. You can see here that this radius is given by the distance between f(x) and the x-axis for any disk along the x-axis.

So once I have the volume of a single disk, I can find the volume of the entire football by adding up all of the volumes of all of the disks. That is the sum, over all of the disks, of the volume of each disk, or the sum from k=1 (disk 1) to disk n of pi times the radius of the disk squared (pi *(r sub k)^2) times the thickness of the disk (delta * x sub k). As I take an infinite number of disks, each having an infinitely thin thickness, I end up not only with a great estimate of the total volume, but with an integral, which is just a Riemann sum. It is the integral from the left side, a, to the right side, b, of pi (r(x)^2) * dx.

The Disk Method

So let's do this for our football, where we're going from x=0 on the left side to x=4 on the right side. My volume is 0 to 4 of pi times the radius squared, which is 2-(1/2)(x-2)^2, all squared, times dx. If I calculate this, I'll end up with the volume of my football. That's not just the area of the region. Because I've got pi and I'm squaring the radius, I've got the entire region that's generated when I spin it along the x-axis. This is what's known as the disk method for volumes of revolution. In this method, we calculate the volume as being the integral from x=a to x=b of pi times the radius squared at x. You can picture this just by keeping in mind your generating region and your axis of rotation. So you're taking your generating region and you're rotating it around, in this case the x-axis, to end up with some sort of cylindrical figure like your football.

The sum of the volumes of all the disks gives the volume of the entire football
Whole Football Volume

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