Answer to Question #268876 in Calculus for Bhuvana

Question

Use a triple integral to find the volume of the wedge in the first octant that is cut from the solid cylinder y

2 + z

2 ≤ 1

by the planes y = x and x = 0

Answer 1

The wedge can be described as the region:

$D=\{(x,y,z)|0\le z^2\le \sqrt{1-y^2},0\le y\le 1,0\le x\le y\}$

$V=\int^1_0\int^y_0\int^{\sqrt{1-y^2}}_0 dzdxdy$

in cylindrical coordinates:

$y=rcos\theta,z=rsin\theta,x=x$

$V=\int^{\pi/2}_0\int^1_0\int^{rcos\theta}_0 dxrdrd\theta=\int^{\pi/2}_0\int^1_0r^2cos\theta drd\theta=$

$=\int^{\pi/2}_0\frac{cos\theta}{3}d\theta=1/3$