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
The wedge can be described as the region:
D={(x,y,z)∣0≤z2≤1−y2,0≤y≤1,0≤x≤y}D=\{(x,y,z)|0\le z^2\le \sqrt{1-y^2},0\le y\le 1,0\le x\le y\}D={(x,y,z)∣0≤z2≤1−y2,0≤y≤1,0≤x≤y}
V=∫01∫0y∫01−y2dzdxdyV=\int^1_0\int^y_0\int^{\sqrt{1-y^2}}_0 dzdxdyV=∫01∫0y∫01−y2dzdxdy
in cylindrical coordinates:
y=rcosθ,z=rsinθ,x=xy=rcos\theta,z=rsin\theta,x=xy=rcosθ,z=rsinθ,x=x
V=∫0π/2∫01∫0rcosθdxrdrdθ=∫0π/2∫01r2cosθdrdθ=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=V=∫0π/2∫01∫0rcosθdxrdrdθ=∫0π/2∫01r2cosθdrdθ=
=∫0π/2cosθ3dθ=1/3=\int^{\pi/2}_0\frac{cos\theta}{3}d\theta=1/3=∫0π/23cosθdθ=1/3
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