Question

Evaluate the line integral, where C is the given curve. C (x + yz) dx + 2x dy + xyz dz, C consists of line segments (2, 0, 1) to (3, 2, 1) and from (3, 2, 1) to (3, 4, 4)

Answer 1

Parameterize the first line segment by

$\vec r(t)=(1-t)(2,0,1)+t(3,2,1)=(2+t,2t,1)$

and the second by

$\vec s(t)=(1-t)(3,2,1)+t(3,4,4)=(3,2+2t,1+3t)$

both with $0\le t\le1$. Then

$\displaystyle\int_C(x+yz)\,\mathrm dx+2x\,\mathrm dy+xyz\,\mathrm dz$

$=\displaystyle\int_0^1\bigg(((2+t)+2t)(1)+2(2+t)(2)+(2+t)(2t)(1)(0)\bigg)\,\mathrm dt$

$\displaystyle\quad+\int_0^1\bigg((3+(2+2t)(1+3t))(0)+2(3)(2)+3(2+2t)(1+3t)(3)\bigg)\,\mathrm dt$

$=\displaystyle\int_0^1(54t^2+79t+40)\,\mathrm dt=\boxed{\frac{195}2}$