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I've got the function y is equal to X minus 3 squared times X minus 1 and what I want to do is think about rotating the part of this function that sits right over here between X is equal to 1 and x equals 3 and x equals 3 and x equals 1 or clearly the zeros of this function right over here and I want to take this region I want to take this region and rotate it around the y axis and if I did that I get a shape that looks something like that and I want to figure out the volume of that shape and what we're going to do is a new method called the shell the shell method and the reason where we use the shell method you might say hey in the past we've rotated things around a vertical line before we use the disk method we wrote everything as a function of Y etc etc we created all of these disks we figured out the volume of each of those disks but the problem here is this is hard to express as a function of Y it's hard to express how do you solve explicitly for Y right over here so instead we're going to keep things in terms of X and have a different geometric visualization for how we can come up with the volume what we're going to imagine instead instead of constructing disks we're going to construct shells and what do I mean by a shell so for each exit the interval we can say we can construct on this kind of cut of it we can construct a rectangle and what happens if we were to rotate this rectangle so this is the rectangle right over here this is the rectangle right over here what happens if we rotate this rectangle around the y axis along with everything else well it's going to look something like I'll try my best attempt to draw it it's going to look something like something like this this is challenging my art skills but I think I can handle it so it's going to look something something not too dissimilar to that right over there so it looks kind of like a hollowed-out cylinder or I guess that's what we call it a shell and it's going to have some depth the depth is going to be DX so the depth right over here is going to be DX the depth right over here is going to be DX and the height right over here is going to be the value of my function the height is f of X in this case f X is X minus 3 squared times X minus 1 how do we figure out the volume of a cylinder like this well if we can figure out the circumference of the cylinder if we can figure out the circumference of the cylinder and then multiply that circumference times the height of the cylinder we'd essentially figure out the area of the outside surface of our cylinder and then if we multiply the area of the outside surface of our cylinder by that infinitesimally small depth then that will give us the volume I shouldn't say cylinder of our shell so let's try to do it what is the circumference what is the circumference of a shell the circumference circumference what is the circumference of one shell going to be well it's going to be 2 pi times the radius of that shell which is going to be we need to express this as a function of X and so what is that going to be it's going to be two pi so for a given x what is the radius well the radius right over here is just the distance the horizontal distance between the y-axis and that X and that's so that's just X so the circumference in this case it's just going to be 2 pi times X now what is the height going to be at any one of those at any one of those for any one of those shells the height is going to be f of X f of X this is right over here and so what is going to be the surface area of the outside so let me put this in quotes outside outside surface area surface area I'm not well I'm not worried about the depth right now the DX I'm not worried about this kind of top part and the bottom part I'm just worried about the outside surface area well the outside surface area is just going to be the circumference times the height it's going to be 2 pi x times f of X times f of X and in this situation in the situation we're looking at right over here that's going to be 2 pi X times X minus 3 squared times X minus 1 now what's going to be the volume so the volume of the shell shell shell volume is just to be all of this business x times DX so it's going to be 2 pi x times f of X DX and so now we're ready to integrate we're ready to integrate over the interval so the volume of our entire shape the volume of our entire shape is going to be the definite integral we're going to integrate over all the X's in the interval from X is equal to 1 to X is equal to 3 X is equal to 1/2 x equals 3 of this thing and we could take the 2 pi out front so we'll put 2 pi out front and on the inside we have x times f of X which in our situation is this business so it's going to be x times X minus 3 squared times X minus 1 and then of course we have our we have our DX so there you have it using the shell method we have set up our definite integral for the volume of this strange-looking shape right over there