Shell method for rotating around horizontal line

what we're going to do in this video is take the function y is equal to the cube root of x and then rotate this around the x-axis and if we do that we get a solid of revolution that looks like that and we're doing it between X is equal to 0 and X is equal to 8 and you get something that looks like this and you could find the volume of this actually quite easily using the disk method but just to show you that you can do it an alternate way we're going to use the shell method but we're going to use the shell method now to rotate around a horizontal line and specifically the x axis so how would we do this well what we want to do what you could imagine is constructing rectangles constructing a rectangle that looks like this let me do it in this salmon color so you have a rectangle that looks something like that its depth or type you could say is dy and then its length right over here is going to be 8 minus whatever x value this 4 is let me make this clear the length the the width the width is going to be 8 8 minus whatever x value this is right over here and you might already realize that this is going to be dy we're going to be integrating with respect to Y over an interval over an interval in Y and so we really want to have everything in terms of Y so this x value whatever it is as a function of Y so if Y is equal to the cube root of x we can cube both sides and we can get X is equal to Y to the third power these are equivalent statements right over here and so this distance right over here is going to be 8 minus y to the 3rd if we were to express it in terms of Y and when you rotate that thing around the x-axis it's going to construct the outside of a cylinder or a shell as we like to call it and I'll try my best to draw that shell so it will look something like this it will look something like so you're going to have a shell you're going to have a shell that looks that looks like this and there you go and then actually this shares a common boundary right over here so you're going to have a shell that looks something like that so that hopefully helps a little bit let me give it some depth so that let me give it you'll have a shell that looks something like that is if we could figure out the volume of that shell which is really going to be the volume of the outer surface area times or the area of the outer surface area times the depth and then if we were to sum up all of those shells this this shell is for a particular Y in our interval if we were to sum up over all of the Y's interval all of the shells all the volumes of the shells then we have the volume of this figure so once again how do we figure out the volume of a shell well we can figure out we can figure out the circumference of kind of the top or I guess the left or the right in this case the left or the right of our cylinder if we can figure out that circumference that circumference circumference is equal to two pi times the radius and what is the radius the radius of these things are just going to be your Y value that's this distance right over here that's just going to be the Y value so it's going to be equal to 2 pi times y and then if we want the area of the outer surface we just multiply that the circumference times the width times the width of our cylinder times the width of our cylinder so let me write it over here so outer surface area outer surface surface area is going to be equal to our 2 pi Y 2 pi Y times our 8 minus y to the third 8 minus y to the third 8 minus 1/3 is just this length you multiply that times circumference you get the outer surface area now if you want to find the volume of this of this one shell it's going to be the outer surface area 2 pi Y times 8 minus y to the third times the depth times this dy right over here times the dy I'll do the dy in purple times the dy times the dy so that's the volume of one shell if we want to find the volume of the entire solid or revolution we have to sum all of these up and then take the limit as they become infinitely thin and we have an infinite number of these shells so we're going to take the sum from and remember we're dealing in Y so the volume the volume is going to be equal to so what's our interval in terms of Y so Y definitely starts off at 0 and when X is equal to 8 what is y well X 8 to the 1/3 power is just 2 so Y is 2 this value right over here let me make it a little bit clearer this value right over here is 2 so Y goes from 0 to 2 and we set up our integral and this one looks pretty straightforward so I think we can crank through it in this video so this is going to be equal to we can take out the 2 pi 2 pi times the definite integral from 0 to 2 of let's multiply the Y of 8y minus y to the 4th all of that dy all of that dy this is going to be equal to 2 pi times the antiderivative of this business antiderivative of 8y is 4 y squared 4 y squared antiderivative of 1 negative Y to the fourth is negative Y to the fifth over 5 and we're going to evaluate it at 0 and 2 so this is going to be equal to 2 pi times you evaluate this business at 2 2 squared is 4 times 4 is 16 16 minus 2 to the fifth 2 to the fifth is 32 so minus 32 over 5 minus 32 over 5 and then you evaluate this stuff at 0 you just get 0 so that's what we're left with and now we just have to simplify this thing a little bit so let's see 16 over 5 this part right over here 16 over 5 is the same thing as 80 or 16 is the same thing I should say is 80 over 5 and from that we're subtracting 32 over 5 so I'm from that we're subtracting 32 over 5 and so that is equal to 48 over 5 so all of this business is equal to 48 over 5 did I do that right 80 minus 30 is 50 and then - another 2 gets us to 48 so this 48 over 5 times 2 pi and now we deserve our drumroll 48 times 2 is 96 under this in a new color just to emphasize that we're at the end 48 times 2 is 96 times pi over 5 so once again this is something that you could have solved using the desk disk method in terms of X and we're just showing that you could also term it to solve it in the shell method in terms of Y the shell or the the hollowed-out cylinder method whatever you want to call it in terms of y