Question

Find the area of the helicoid (or spiral ramp) with vector equation r(u, v) = ucos(v) i + usin(v) j + v k, 0 ? u ? 1, 0 ? v ? 9?.

Answer 1

The area is given by the surface integral


`\displaystyle\iint_S\mathrm dS=\int_(u=0)^(u=1)\int_(v=0)^(v=9)\left\|\mathbf r_u*\mathbf r_v\right\|\,\mathrm dv\,\mathrm du`

`=\displaystyle\int_(u=0)^(u=1)\int_(v=0)^(v=9)√(u^2+1)\,\mathrm dv\,\mathrm du`

`=\frac9{\sqrt2}+\frac92\ln(\sqrt2+1)`