Question

Find the area of the surface. the part of the surface z = xy that lies within the cylinder x2 y2 = 25.

Answer 1

Parameterize the intersection by


`\mathbf r(r,\theta)=(r\cos\theta,r\sin\theta,r^2\cos\theta\sin\theta)`

with
`0\le r\le5` and
`0\le\theta\le2\pi`. Then the area is given by the surface integral


`\displaystyle\int_(r=0)^(r=5)\int_(\theta=0)^(\theta=2\pi)\left\|\mathbf r_r*\mathbf r_\theta\right\|\,\mathrm d\theta\,\mathrm dr`

`\displaystyle\int_0^(2\pi)\int_0^5r√(1+r^2)\,\mathrm dr\,\mathrm d\theta`

`\displaystyle\pi\int_1^(26)\sqrt s\,\mathrm ds`

`\displaystyle\frac{2\pi}3s^(3/2)\bigg|_(s=1)^(s=26)=\frac{2\pi(26^(3/2)-1)}3`