Shell method with two functions of x

this right here is a solid or ever solid of revolution whose volume we were able to figure out in previous videos actually in a different tutorial using the disk method and integrating in terms of why we're going to do it now is we're going to find the same volume for the same solid of revolution but we're going to do it using the shell method and integrating with respect to X so what we do is we have the region between these two curves Y is equal to square root of x and y is equal to x squared and we're going to rotate it around the vertical line x equals two and we're going to do that and at the interval that we're going to rotate this space between these two curves is the interval between when square root of x is greater than x squared and so it's between zero and one and so let's try to do it with the shell method and so to do that what we do is we want to construct a shell so let's imagine let me just in a different color let's imagine a rectangle right over here let's imagine a rectangle right over there it has width DX so it has width DX and its height is the difference of these two functions and so if I were to draw it right over here it would look something like this so it would look something like that so it'd be there it'd be there and then it is a shell it's kind of a hollowed-out cylinder so it would look something something like this it would look something like that just like that and it has some depth that's what the DX gives us so we have the depth that looks something like that and then let me shade it in a little bit just so we can see a little bit of its see a little bit of its depth so when you rotate this rectangle around the line x equals two you get a shell like this so let's think about how we can figure out the volume of this shell well we've already done this several times the first thing we might want to think about is the circumference of kind of the top of the shell the circumference of the top of the shell we know circumference is 2 pi times radius we just need to know what the radius of the shell is what is this distance what is that distance going to be well it's the horizontal distance between X equals two and whatever the x-value is right over here so it's going to be it's going to be two minus our x-values so this radius this distance right over here is going to be two minus X and so the circumference is going to be that times two pi two pi R gives us the circumference of that circle so 2 pi times two minus X and then if we want the surface area of the outside of our shell so the area is going to be the circumference 2 pi times 2 minus x times the height of each shell now what is the height of each shell well it's going to be the distance between it's going to be the vertical distance expressed as functions of Y so it's going to be the top boundary is y is equal to square root of x the bottom boundary is y is equal to x squared so it's going to be square root of x minus x squared so let me make this so it's going to be it's going to be let me do this in the yellow so it's going to be square root of x minus minus x squared and so if you want the volume of a given shell it's going to be and I'll write all of this in white it's going to be 2 pi times 2 minus x times square root of x minus x squared so this whole expression we just I just rewrote it is the SIRT is e is the area the outside surface area of one of these shells if we want the volume we have to give it a little bit of depth multiplied by how deep the shell is so times DX and if we want the volume of this whole thing we just have to sum up all the shells for all of the X's in this interval and take the limit as the DX is gets smaller and smaller and we have more and more shells and so what's our interval well our X's are going to go between 0 & 1 0 & 1 so that right over there is the volume of this is the volume of this figure