Answer to Question #196540 in Calculus for Jean Lyre de Leon

Question

Find the volume generated if the function y= square root of 1-x^2 is revolved around the x-axis (use only the positive area)

Answer 1

The volume of Revolution about $O_x$ is given by:

$V = \int^{x=a}_{x=b}\pi y^2dx$

So for this problem, Nothing that $1-x^2 = 0 \implies x = \pm 1$ and that by symmetry we can double the volume for the region but we have to consider only positive coordinate.

$V = \int_{0}^{3} \pi(\sqrt{1-x^2})^2dx$

$V = \pi \int_{0}^{3}(1-x^2)dx$

$V = \pi[x-\dfrac{x^3}{3}]_{0}^{1}$

$V = \dfrac{2\pi}{3}$