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Citrus2011 [14]
3 years ago
10

Find the area of the surface. The part of the paraboloid z = x2 + y2 that lies inside the cylinder x2 + y2 = 9.

Mathematics
1 answer:
Sloan [31]3 years ago
5 0
Parameterize the surface S by

\mathbf s(r,\theta)=(x(r,\theta),y(r,\theta),z(r,\theta))=(r\cos\theta,r\sin\theta,r^2)

with 0\le r\le3 and 0\le\theta\le2\pi.

The area of S is then given by the surface integral

\displaystyle\iint_S\mathrm dS=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}\left\|\mathbf s_r\times\mathbf s_\theta\right\|\,\mathrm dr\,\mathrm d\theta
=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}r\sqrt{1+4r^2}\,\mathrm dr\,\mathrm d\theta
=\displaystyle\dfrac\pi6(1+4r^2)^{3/2}\bigg|_{r=0}^{r=3}
=\dfrac{(37\sqrt{37}-1)\pi}6
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