Question

Find the area of the surface. the part of the cylinder y2 + z2 = 9 that lies above the rectangle with vertices (0, 0), (5, 0), (0, 2), and (5, 2)

Answer 1

Same parameterization as in the provided link, which I'll reproduce correctly here:

$\mathbf s(u,v)=\langle u,3\cos v,3\sin v\rangle$

and this time $0\le u\le5$ and $\cos^{-1}\dfrac23\le v\le\dfrac\pi2$

We get the same surface element,

$\mathrm d\mathbf S=3\,\mathrm du\,\mathrm dv$

and the area is

$\displaystyle3\iint_{v=\cos^{-1}(2/3)}^{v=\pi/2}\int_{u=0}^{u=5}\mathrm du\,\mathrm dv=\dfrac{15\pi}2-15\cos^{-1}\dfrac23$