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

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

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

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

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

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

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