Question

Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. modifyingbelow integral from nothing to nothing with upper c xyz font size decreased by 5 ds​; c is the line segment from left parenthesis 0 comma 0 comma 0 right parenthesis to left parenthesis 1 comma 3 comma 6 right parenthesis

Answer 1

Parameterize $C$ by

$\vec r(t)=(1-t)(0,0,0)+t(1,3,6)=(t,3t,6t)$

with $0\le t\le1$. Then the line integral is

$\displaystyle\int_Cxyz\,\mathrm dS=\int_0^1x(t)y(t)z(t)\left\|\frac{\mathrm d\vec r}{\mathrm dt}\right\|\,\mathrm dt$

$=\displaystyle18\sqrt{91}\int_0^1t^3\,\mathrm dt=\boxed{\frac{9\sqrt{91}}2}$