Question

Use stokes' theorem to evaluate c f · dr where c is oriented counterclockwise as viewed from above. f(x, y, z = xyi + 5zj + 7yk, c is the curve of intersection of the plane x + z = 5 and the cylinder x2 + y2 = 36.

Answer 1

The intersection can be parameterized by


`C:=\mathbf r(t)=\begin{cases}x(t)=6\cos t\\y(t)=6\sin t\\z(t)=5-6\cos t\end{cases}`

with
`0\le t<2\pi`.

By Stoke's theorem, the integral of
`\mathbf f(x,y,z)=xy\,\mathbf i+5z\,\mathbf j+7y\,\mathbf k` along
`C` is equivalent to


`\displaystyle\int_C\mathbf f(x(t),y(t),z(t))\cdot\mathrm d\mathbf r(t)=\iint_S\\abla*\mathbf f\,\mathrm dS`

where
`S` is the region bounded by
`C`. The line integral reduces to


`\displaystyle\int_0^(2\pi)(36\sin t\cos t\,\mathbf i+(25-30\cos t)\,\mathbf j+42\sin t\,\mathbf k)\cdot(-6\sin t\,\mathbf i+6\cos t\,\mathbf j+6\sin t\,\mathbf k)\,\mathrm dt`

`=\displaystyle\int_0^(2\pi)(54(\cos3t-\cos t)-30(3\cos2t-5\cos t+3)+(126-126\cos2t)\,\mathrm dt`

`=\displaystyle\int_0^(2\pi)(36+96\cos t-216\cos2t+54\cos3t)\,\mathrm dt`

`=72\pi`