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Aleksandr [31]
3 years ago
9

Find the area of the surface. the part of the hyperbolic paraboloid z = y2 − x2 that lies between the cylinders x2 + y2 = 4 and

x2 + y2 = 16.g
Mathematics
1 answer:
Nadusha1986 [10]3 years ago
8 0
Call the surface S; then the area of S is given by the surface integral

\displaystyle\iint_S\mathrm d\mathbf S

Parameterize the surface by

\mathbf r(u,v)=\begin{cases}x(u,v)=u\cos v\\y(u,v)=u\sin v\\z(u,v)=u^2\cos^2v-u^2\sin^2v=u^2\cos2v\end{cases}

with 2\le u\le4 and 0\le v\le2\pi. The surface element is given by

\mathrm d\mathbf S=\|\mathbf r_u\times\mathbf r_v\|\,\mathrm du\,\mathrm dv
\mathrm d\mathbf S=u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv

So the area is

\displaystyle\iint_S\mathrm d\mathbf S=\int_{v=0}^{v=2\pi}\int_{u=2}^{u=4}u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv
=\displaystyle2\pi\int_{w=17}^{w=65}\sqrt w\,\frac{\mathrm dw}8

where w=1+4u^2\implies\mathrm dw=8u\,\mathrm du

=\displaystyle\frac\pi4\frac23w^{3/2}\bigg|_{w=17}^{w=65}
=\dfrac\pi6\left(65^{3/2}-17^{3/2}\right)\approx237.69
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