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 + 2 z = 5 inside the cylinder x^2 + y^2 = 1.

Answer 1

Parameterize the surface by


`\mathbf s(u,v)=\left\langle u\cos v,u\sin v,\frac{5-u\sin v}2\right\rangle`

with
`0\le u\le1` and
`0\le v\le2\pi`. Then the surface element is


`\mathrm dS=\|\mathbf s_u*\mathbf s_v\|\,\mathrm du\,\mathrm dv=\frac{\sqrt5}2u\,\mathrm du\,\mathrm dv`

and the area of the surface
`\mathcal S` is given by


`\displaystyle\iint_(\mathcal S)\mathrm dS=\frac{\sqrt5}2\int_(v=0)^(v=2\pi)\int_(u=0)^(u=1)u\,\mathrm du\,\mathrm dv=\frac{\pi\sqrt5}2`