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lbvjy [14]
3 years ago
8

Evaluate the surface integral. ??s (x2 y2 z2 ds s is the part of the cylinder x2 y2 = 4 that lies between the planes z = 0 and z

= 2, together with its top and bottom disks.
Mathematics
1 answer:
kondor19780726 [428]3 years ago
6 0
The surface S can be parameterized by

\mathbf r(\theta,z)=(2\cos\theta,2\sin\theta,z)

where 0\le\theta\le2\pi and 0\le z\le2. Then the surface integral can be computed with

\displaystyle\iint_S(x^2+y^2+z^2)\,\mathrm dS
\displaystyle=\int_{\theta=0}^{\theta=2\pi}\int_{z=0}^{z=2}(4\cos^2\theta+4\sin^2\theta+z^2)\|\mathbf r_\theta\times\mathbf r_z\|\,\mathrm dz\,\mathrm d\theta=\dfrac{128\pi}3
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