Help on a calculus problem.

Question

Use the Shell Method to compute the volume of the solids obtained by rotating the region enclosed by the graphs of the functions y=x^2, y=8−x^2 and to the right of x=1 about the y-axis.

Answer 1

First, graph the region.

 

The curves intersect when 8-x² = x².  So, they intersect when x = 2.

 

The region lies in quadrant 1.  It is bounded on the left by the vertical line x = 1, on the right by the vertical line x = 2, above by the curve      

y = 8-x², and below by y = x².

 

Take a typical vertical slice of the region of width Δx at a distance x from the y-axis.

 

Rotate the slice about the y-axis.  The resulting shell has volume         2π(radius)(height)(thickness).

 

 radius = x

 height = (8-x²)-x² = 8-2x²

 thickness = Δx

 

Volume of shell = 2πx(8-2x²)Δx = 4π(4x-x³)Δx

 

Volume of solid = 4π∫_{(from x=1 to x=2)}(4x-x³)dx