Question

Find the volume of the solid right cylinder whose base is the region in the xy – plane that lies inside the cardioid r = 1 + cos θ and outside the circle r = 1 and whose top lies in the plane z = 4.

Answer 1

The cylinder's volume is the product of its height and the area of its base. We're told that the height is restricted to the plane $z=4$, so the height is 4.

The area of the base $B$ is obtained by the integral,

$\displaystyle\iint_B\mathrm dx\,\mathrm dy=\int_{-\pi/2}^{\pi/2}\int_1^{1+\cos\theta}r\,\mathrm dr\,\mathrm d\theta=\frac{8+\pi}4$

Then the volume of the cylinder is 8 + π.