Answer to Question #268876 in Calculus for Bhuvana

Question #268876

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


1
Expert's answer
2021-11-24T16:09:42-0500

The wedge can be described as the region:

D={(x,y,z)0z21y2,0y1,0xy}D=\{(x,y,z)|0\le z^2\le \sqrt{1-y^2},0\le y\le 1,0\le x\le y\}


V=010y01y2dzdxdyV=\int^1_0\int^y_0\int^{\sqrt{1-y^2}}_0 dzdxdy


in cylindrical coordinates:

y=rcosθ,z=rsinθ,x=xy=rcos\theta,z=rsin\theta,x=x


V=0π/2010rcosθdxrdrdθ=0π/201r2cosθdrdθ=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=


=0π/2cosθ3dθ=1/3=\int^{\pi/2}_0\frac{cos\theta}{3}d\theta=1/3


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