Question

can anyone help me with the last problem # 14

Find the volume of the given solid generated by revolving the regions bounded by the graphs of the equations about the given lines.

14. y=6-2x-x^2, y=x+6
a. the x-axis
b. the line y=3

Answer 1

The graphs intersect at (-3,3) and (0,6)

Using vertical strips is easiest, since the upper and lower boundaries do not change. That means we integrate on x.

For revolving around the x-axis, that means using discs (washers) of thickness dx, so

v = ∫[-3,0] π(R^2-r^2) dx
where R=6-2x-x^2 and r=x+6
v = ∫[-3,0] π((6-2x-x^2)^2-(x+6)^2) dx = 243π/5

To revolve around y=3, the same idea is used, but each radius is now y-3 instead of y.