Answer to Question #289096 in Calculus for jea

Question #289096

Find the volume of the solid S enclosed by the cylinder x2 + y2 = 9 and the planes y + z = 5 and

 = 1. (Hint: Convert to Cylindrical/Polar Coordinates)


1
Expert's answer
2022-01-25T12:29:51-0500
V=dV=339x29x215ydzdydxV=\int \int \int dV=\displaystyle\int_{-3}^{3}\displaystyle\int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}\displaystyle\int_{1}^{5-y}dzdydx

=339x29x2(4y)dydx=\displaystyle\int_{-3}^{3}\displaystyle\int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}(4-y)dydx

=02π03(4rsinθ)rdrdθ=\displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^{3}(4-r\sin \theta)rdrd\theta

=02π[2r2r33sinθ]30dθ=\displaystyle\int_{0}^{2\pi}[2r^2-\dfrac{r^3}{3}\sin \theta]\begin{matrix} 3 \\ 0 \end{matrix}d\theta

=02π(189sinθ)dθ=\displaystyle\int_{0}^{2\pi}(18-9\sin \theta)d\theta

=[18θ+9cosθ]2π0=36π(units3)=[18\theta+9\cos \theta]\begin{matrix} 2\pi \\ 0 \end{matrix}=36\pi ({units}^3)

36π36\pi cubic units.


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