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 4, 8, 32 3 .

Answer 1

Let $x=t$, so that

$x^2=2y\implies t^2=2y\implies y=\dfrac{t^2}2$

$3z=xy\implies3z=\dfrac{t^3}2\impiles z=\dfrac{t^3}6$

Then the length of the path is

$\displaystyle\int_{t=0}^{t=4}\sqrt{\left(\frac{\mathrm dx}{\mathrm dt}\right)^2+\left(\frac{\mathrm dy}{\mathrm dt}\right)^2+\left(\frac{\mathrm dz}{\mathrm dt}\right)^2}\,\mathrm dt$
$=\displaystyle\int_0^4\sqrt{1+t^2+\frac{t^4}4}\,\mathrm dt$
$=\displaystyle\frac12\int_0^4\sqrt{4+4t^2+t^4}\,\mathrm dt$
$=\displaystyle\frac12\int_0^4\sqrt{(t^2+2)^2}\,\mathrm dt$
$=\displaystyle\frac12\int_0^4(t^2+2)\,\mathrm dt$
$=\dfrac{44}3$