Chapter 16 - Section 16.2 - Line Integrals - 16.2 Exercise - Page 1142: 32

$-2$

$\int_{c} F(x,y,z)$, $F(x,y,z) = xi -z j + yk$ where $C$ is $r(t)=2ti + 3tj -t^2 k$ $ r'(t) = 2i +3 j -2tk$ $$\int_{c} F. dr = \int_{a}^b F(r(t)). r'(t) dt$$ $ = \displaystyle\int_{-1}^1 (2ti+ t^2j +3t k). ( 2i +3 j -2tk) dt$ $ = \displaystyle\int_{-1}^1 (4t+ 3t^2- 6t^2) dt= \int_{-1}^1 (4t -3t^2)dt$ $=(2t^2-t^3)| _{-1}^1 = -2$