Generalizing the disc method around x-axis
About this video
The video generalises the previous example to any function y = f(x) between x = a and x = b rotated around the x-axis. Each disk has radius f(x), face area pi f(x)^2 and depth dx, so the volume is the integral from a to b of pi f(x)^2 dx, the textbook disk method formula.
You'll be able to
- explain where the disk method formula V = integral from a to b of pi f(x)^2 dx comes from.
- identify f(x) as the radius of each disk when rotating around the x-axis.
- describe the volume as a limit of a sum of infinitely thin disks.
Key words
- disk method
- Finding the volume of a solid of revolution by adding up the volumes of thin disks, using an integral.
- radius
- The distance from the axis to the edge of a disk; around the x-axis it equals f(x).
- first principles
- Working something out from the basic idea (area times depth, then sum) rather than from a memorised formula.
Transcript
What we're going to do in this video is generalize what we did in the last video. And, essentially, end up with the formula for rotating something around the X-axis like this using, what we call, the "disk method". And the point is to show you where that formula inside a calculus textbook actually comes from.
But it just comes from the same exact principles we did in the last video! It's not advised to memorize the formula; I highly recommend against that because you really need to know what's going on. It's really better to do it in first principles where you find the volume of each of these disks.
But let's just generalize what we saw in the last video. So instead of saying that "y = x^2", let's just say that this is the graph, the function that's right over here. Let's just generalize it and call it "y=f(x)". And instead of saying x is going from 0 to 2 let's say that we're going between a and b, so these are just two endpoints along the X-axis.
So how would we find the volume of this? Well, just like the last video, we still take a disk just like this. But what is the height of the disk? The height of the disk is not just x^2 since we've generalized it. So the height is simply going to be whatever the height of the function is at that point.
So the height of the disk is going to be f(x)! The area of the space of this disk is going to be πR^2. So our radius is f(x), and we're just going to square it. That's the area, that's the area of this face right over here. What is the volume of the disk? We're just going to multiply that by our depth, which going to be dx.
And we want to take the sum of all of these disks from a to b, and we're going to take the sum of them, and we're going to take the limit as the "dx"s get smaller and smaller. And we have an infinite number of these disks. Thus, we are going to the integral of this from a to b!
And this right here is the formula that you will see often in a calculus textbook for using the disk method as you're rotating around the X-axis. So I just wanted to show you that it comes out of the common sense of finding the volume of this disk. The f(x) right over here is just the radius of the disk, so this part over here is really just πR^2; we multiply it [area of the circle] times the depth; then we take the sum from a to b of all of the disks.
And essentially since this is integral is the limit of all the disks getting narrower and narrower. And thus, we have an infinite number of these disks.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.