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Current time:0:00Total duration:2:35

AP.CALC:

CHA‑5 (EU)

, CHA‑5.C (LO)

, CHA‑5.C.1 (EK)

what we're going to do in this video is generalize what we did in the last video and essentially end up with a formula for rotating something around the x axis like this using what we would call the disk method and the point is to really show you where that formula that you might see in a calculus book actually comes from that it just comes from the exact same principles that we did in the last video it's not for you to memorize it formula I highly recommend against that because then you really won't know what's going on it's really just to do it it's better to do it from first principles where you find the volume of each of these discs and think of it that way but let's just generalize what we saw in the last video so instead of saying that this is y is equal to x squared let's say that this is the graph this function right over here this function right over here let's just generalize it and call it Y is equal to f of X and instead of saying but we're going between 0 & 2 let's just say we're going between 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 would still take a disk just like this but what is the height of the disk now well the height of the disk is not just x squared we've generalized it it is going to be whatever the value of our function is at that point so the height of the disk is going to be f of X the area of this of the space of this disk is going to be pi times our radius squared so our radius is f of X and we are just going to square it that's the area that's the area of this face right over here what is the volume of our disk we're just have to multiply that times our depth so it's going to be that times 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 then take the limit as the DX is get smaller and smaller and smaller and we have an infinite number of these disks and that means we're just going to take a definite integral so we're going to take the definite integral of this from A to B and this right here is the formula that you will often see in a calculus book for using the disk method when you're rotating around the x axis but I just wanted to show you that it comes out of the common sense of finding the volume of this disk the f of X right over here is just the radius of the disk so this this part right over here is just really PI R squared we multiply it times the depth and then we take the sum from A to B from A to B of all of the discs and it's really essentially since this is an integral it's the limit as each of those discs get narrow narrow narrow and we have an infinite number of those discs

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