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\|(\partial\mathbf x)/(\partial u)*(\partial\mathbf x)/(\partial v)\right\|\,\mathrm dv\,\mathrm du=90√(110)\pi`