Question

Let c be the curve of intersection of the parabolic cylinder x2 = 2y, and the surface 3z = xy. find the exact length of c from the origin to the point 2, 2, 4 3 . step 1

Answer 1

Parameterize the curve
`C` by
`\mathbf r(t)=\left\langle t,\frac{t^2}2,\frac{t^3}6\right\rangle` (essentially replacing
`x=t` and finding equivalent expressions for
`y,z` in terms of
`t`.

The length of
`C` is given by the line integral


`\displaystyle\int_C\mathrm dS=\int_(t=0)^(t=2)\|\mathbf r'(t)\|\,\mathrm dt`

`=\displaystyle\int_0^2\left\|\left\langle1,t,\frac{t^2}2\right\rangle\right\|\,\mathrm dt`

`=\displaystyle\int_0^2\sqrt{1+t^2+\frac{t^4}4}\,\mathrm dt`

`=\displaystyle\frac12\int_0^2√(4+4t^2+t^4)\,\mathrm dt`

`=\displaystyle\frac12\int_0^2√((t^2+2)^2)\,\mathrm dt`

`=\displaystyle\frac12\int_0^2(t^2+2)\,\mathrm dt`

`=\frac12\left(\frac{t^3}3+2t\right)\bigg|_(t=0)^(t=2)`

`=\frac12\left(\frac83+4\right)`

`=\frac{10}3`