I know how to do very basic and simple shell method problems, but this one is difficult for me. Every problem that I have had to do so far, the graphs always stay on one side of the x and y axis. Meaning they never cross it. This one does, however, and I don't understand how to work with that. So the information given is:
x= y^4/4 -y^2/2
x=y^2/2
There is a graph of this shown, and the graphs intersect at (0,0) and (2,2).
But the graph of y^4/4 - y^2/2 goes over to the left side of the y axis for a bit, and then goes back to the right. I need to find the volume of the region that would be formed by rotating the graph about the x axis. So how do I find the radius and the height for this? Maybe I'm making it more complicated than it is...
Yeah, that's what I meant. Sorry I'm a little slow on the reply. So yeah, it's being rotated about the x axis. But in order to use the shell method, I need to know the radius and the height, right? The fact that the graph crosses over to the left side of the y axis is making me unsure of what the radius and height would be.
First you need to define the integration boundaries in x. You can show that it goes from x=-1/4 to x=2
Then you need to separate the calculation into 2 parts :
- from x=-1/4 to x=0 the curve is bounded only by x= y^4/4 -y^2/2. This means that you need to inverse this expression in order to get y function of x.
You will get 2 functions : one (f(x)) for the lower part and one (g(x)) for the upper part
Due to the fact that the rotation is around x axis, the radius is y and the height is x
The volume is
- from x=0 to x=2 the curve is bounded in the lower part by x=y^2/2 and in the upper part by x= y^4/4 -y^2/2. You also need to inverse these expressions to calculate the volume V2