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Use a parametrization to express the area of the surface s as a double integral. then, evaluate the integral to find the area of the surface. s is the portion of the plane y + 3 z = 2 inside the cylinder x^2 + y^2 = 1.

User Aventic
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The cylinder gives you a hint as to where to start with a parameterization
\mathbf s(u,v)=\langle x(u,v),y(u,v),z(u,v)\rangle. A sensible choice would be to set


x=u\cos v

y=u\sin v

so that


y+3z=2\implies z=\frac{2-u\sin v}3

with
0\le u\le1 and
0\le v\le2\pi. Then


\mathbf s_u*\mathbf s_v=\left\langle0,\frac u3,u\right\rangle

so the surface element is


\mathrm dS=\|\mathbf s_u*\mathbf s_v\|\,\mathrm du\,\mathrm dv=\frac{√(10)u}3\,\mathrm du\,\mathrm dv

So the area of the surface is


\displaystyle\iint_(\mathcal S)\mathrm dS=\frac{√(10)}3\int_(v=0)^(v=2\pi)\int_(u=0)^(u=1)u\,\mathrm du\,\mathrm dv=\frac{√(10)\pi}3
User TampaHaze
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