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.

Answer 2

The required volume of the given function is,

$Volume=8+\pi$

The volume of the solid right cylinder:

The formula for finding the volume of the right circular cylinder with the radius $r$ is given by,

$Volume=\frac{h}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(r_{2}^{2}-r_{2}^{1})d\theta$

Now, the given function is,

$r = 1 + cos\theta$

Nows, applying the given function into the above formula we get,

$Area=\frac{1}{2}\int_{\frac{\pi}{2}}^{-\frac{\pi}{2}}{(1+cos\theta)^2-(1)^2}d\theta\\=\frac{1}{2}\int_{\frac{\pi}{2}}^{-\frac{\pi}{2}}(cos^2\theta+2cos\theta)\\=\frac{\pi}{4}+0+2\\=2+\frac{\pi}{4}$

So, the required volume is,

$Volume=4(2+\frac{\pi}{4})\\=8+\pi$

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