Answer to Question #206147 in Calculus for KABIR

Question

• Find the volume of the solid formed by revolving the curve, r=a(1+cosθ), about the initial line.

Answer 1

$V=\displaystyle\int_{0}^{\pi}\dfrac{2}{3}\pi r^2r\sin\theta d\theta$

$=\displaystyle\int_{0}^{\pi}\dfrac{2}{3}\pi (a(1+\cos \theta))^3\sin\theta d\theta$

$=\dfrac{2}{3}\pi a^3\big[-\dfrac{1}{4}(1+\cos \theta)^4\big]\begin{matrix} \pi \\ 0 \end{matrix}$

$=-\dfrac{1}{6}\pi a^3(0-2^4)$

$=\dfrac{8}{3}\pi a^3 (units^3)$

$V=\dfrac{8}{3}\pi a^3 \text{ cubic units}$