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What we're going to do in this video

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is generalize what we did in the last video.

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And, essentially, end up with the formula

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for rotating something around the X-axis like this using,

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what we call, the "disk method".

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And the point is to show you where that formula inside a calculus textbook actually comes from.

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But it just comes from the same exact principles we did in the last video!

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It's not advised to memorize the formula; I highly recommend against that because you really need to know what's going on.

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It's really better to do it in first principles where you find

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the volume of each of these disks.

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But let's just generalize what we saw in the last video.

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So instead of saying that "y = x^2",

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let's just say that this is the graph, the function that's right over here.

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Let's just generalize it and call it "y=f(x)".

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And instead of saying x is going from 0 to 2

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let's say that we're going between a and b, so these are just two endpoints along the X-axis.

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So how would we find the volume of this?

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Well, just like the last video, we still take a disk just like this.

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But what is the height of the disk?

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The height of the disk is not just x^2 since we've generalized it.

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So the height is simply going to be whatever the height of the function is at that point.

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So the height of the disk is going to be f(x)!

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The area of the space of this disk is going to be πR^2.

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So our radius is f(x), and we're just going to square it.

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That's the area, that's the area of this face right over here.

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What is the volume of the disk?

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We're just going to multiply that by our depth, which going to be dx.

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And we want to take the sum of all of these disks from a to b,

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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.

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And we have an infinite number of these disks.

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Thus, we are going to the integral of this from a to b!

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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.

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So I just wanted to show you that it comes out of the common sense of finding the volume of this disk.

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The f(x) right over here is just the radius of the disk, so this part over here is really just πR^2;

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we multiply it [area of the circle] times the depth; then we take the sum from a to b of all of the disks.

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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.