Find the Volume using Cylindrical Shell

Question

The region bounded by y=x^2 and y=sin⁡((pi)(x/2)) is rotated about the line x=−1.

Using cylindrical shells, set up an integral for the volume of the resulting solid.

The limits of integration are:

a= ?

b= ?

and the function to be integrated is: ?

Answer 1

The area of a cross section of a shell is given by:

A(x) = 2(pi)*r*h. The goal is to find functions for r and h from the graph with respect to x, and then integrate A(x) to find the volume.

The radius of any shell within this region is simply given by its x value plus one, since we're rotating about the line x = -1, so r(x) = (x+1).

The height of any shell within this region is the top boundary subtracted from the bottom boundary, so your equation for height is h(x) = sin(pi*x/2) - x^2.

Now, the limits for integration will be given by your two points of intersection. In this case, the two functions intersect at (0,0) and (1,1), so our limits will be 0 and 1.

(SOLUTION) Thus, the limits of integration are a = 0, b = 1, and the function to be integrated is 2(pi)*((x+1)(sin(pi*x/2))-x^2)).