Question

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.

Answer 1

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`