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navik [9.2K]
3 years ago
8

2) Use spherical coordinates to find the integral of f(x,y,z) = z over x^2 + y^2 + z^2 < 4 and x,y,z < 0

Mathematics
1 answer:
777dan777 [17]3 years ago
8 0
1. Denote by \mathcal D the region bounded by the cylinder x^2+y^2=9 and the planes z=0 and z=6. Converting to cylindrical coordinates involves setting


\begin{cases}x=r\cos u\\y=r\sin u\\z=v\end{cases}


and \mathcal D is obtained by varying the parameters over 0\le r\le3, 0\le u\le2\pi, and 0\le v\le6. The volume element is


\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm du\,\mathrm dv

so the integral becomes

\displaystyle\iiint_{\mathcal D}xz\,\mathrm dV=\int_{v=0}^{v=6}\int_{u=0}^{u=2\pi}\int_{r=0}^{r=3}r^2v\cos u\,\mathrm dr\,\mathrm du\,\mathrm dv=0

2. Now let \mathcal D denote the region bounded by the sphere x^2+y^2+z^2=4 and the coordinate planes in the octant in which each coordinate of the point (x,y,z) is negative.

Converting to spherical coordinates, we set

\begin{cases}x=\rho\cos u\sin v\\y=\rho\sin u\sin v\\z=\rho\cos v\end{cases}

where we obtain \mathcal D by varying over 0\le\rho\le2, \pi\le u\le\dfrac{3\pi}2, and \dfrac\pi2\le v\le\pi. The volume element is now

\mathrm dV=\rho^2\sin v\,\mathrm d\rho\,\mathrm du\,\mathrm dv

so the integral becomes

\displaystyle\iiint_{\mathcal D}z\,\mathrm dV=\int_{v=\pi/2}^{v=\pi}\int_{u=\pi}^{u=3\pi/2}\int_{\rho=0}^{\rho=2}\rho^3\cos v\sin v\,\mathrm d\rho\,\mathrm du\,\mathrm dv=-\pi
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