Question

Evaluate the line integral ?Cx2zds, where C is the line segment from (0,5,2) to (2,7,8).

Answer 1

Parameterize $C$ by

$\mathbf r(t)=(x(t),y(t),z(t))=(1-t)(0,5,2)+t(2,7,8)=(2t,5+2t,2+6t)$
$\implies\mathrm ds=\sqrt{\left(\dfrac{\mathrm dx}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dy}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dz}{\mathrm dt}\right)^2}\,\mathrm dt=\sqrt{44}\,\mathrm dt$

So the line integral is equivalent to

$\displaystyle\int_Cx^2z\,\mathrm ds=\sqrt{44}\int_{t=0}^{t=1}(2t)^2(2+6t)\,\mathrm dt=\frac{52\sqrt{11}}3$