Consider the region R in Quadrant I enclosed by the x-axis and y=-2x^2+4x.

Consider V1, V2, and V3, formed by revolving R about y=4, y=-2, and x=4 resp. Compute these volumes.

Here is what I got:

For V1: use washers with outer radius 4 and inner radius 4-(-2x^2+4x). Do pi*integral (0 to 2) 4^2-(4-4x+2x^2)^2 dx, and I got 256*pi/15

For V2: washers, outer of 2+(4x-2x^2), inner of 2. Do pi*integral (0 to 2) ((2+3x-2x^2)^2-4 dx, and I got 224*pi/15

For V3: use shells, radius is 4-x and height is f(x). Do 2pi*integral (0 to 2) of (4-x)*(-2x^2+4x) dx, and I got 16*pi.

Are these answers correct? Thanks.