Question

Find the surface area of that is part of the plane and that lies inside the elliptic cylinder? Plane: 10x+3y+z=10 Cylinder: (x^2)/81+(y^2)/100 = 1

Answer 1

Parameterize the surface of interest $S$ by $\mathbf x(u,v)=\langle9u\cos v,10u\sin v,10-90u\cos v-30u\sin v\rangle$ with $u\in[0,1]$ and $v\in[0,2\pi]$.

Then the area is given by the surface integral
$\displaystyle\iint_S\mathrm dA=\int_0^1\int_0^{2\pi}\left\|\frac{\partial\mathbf x}{\partial u}\times\frac{\partial\mathbf x}{\partial v}\right\|\,\mathrm dv\,\mathrm du=90\sqrt{110}\pi$