Question

Find the surface area of that part of the plane 9x 5y z=5 that lies inside the elliptic cylinder x216 y281=1

Answer 1

I assume the equation of the plane is $9x+5y+z=5$ and the cylinder has equation $\dfrac{x^2}{16}+\dfrac{y^2}{81}=1$.

I don't know what techniques are available to you, so I'll resort to (in my opinion) the most reliable: surface integration.

The surface of intersection $S$ is an ellipse in three dimensional space which can be parameterized by $\mathbf x(u,v)=\langle 4u\cos v,9u\sin v,5-36u\cos v-45 u\sin v\rangle$, with $u\in[0,1]$ and $v\in[0,2\pi]$.

The area is then given by the integral
$\displaystyle\iint_S\mathrm dA=\int_0^1\int_0^{2\pi}\left\|\dfrac{\partial\mathbf x}{\partial u}\times\dfrac{\partial\mathbf x}{\partial v}\right\|\,\mathrm dv\,\mathrm du=36\sqrt{107}\pi$