Question

Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x2 + y2 = 4 and the plane x + y + z = 9.

Answer 1

$4x^2+y^2=4\iff x^2+\left(\dfrac y2\right)^2=1$
$x+y+z=9\iff z=9-x-y$

Let $x=\cos t$ and $y=2\sin t$. Then the cylinder is given when we take $0\le t\le2\pi$.

The cylinder's equation is independent of $z$, which means the intersection is completely determined by the value of $z$ in the plane equation. So the intersection can be described by the vector function $\mathbf r(t)=\langle\cos t,2\sin t,9-\cos t-2\sin t\rangle$.

The length of the intersection curve $C$ is given by the line integral

$\displaystyle\int_C\mathrm d\mathbf r=\int_{t=0}^{t=2\pi}\|\mathbf r'(t)\|\,\mathrm dt$
$=\displaystyle\int_0^{2\pi}\sqrt{\cos^2t+4\sin^2t+(9-\cos t-2\sin t)^2}\,\mathrm dt$
$\approx57.4449$