Chapter 15 - Section 15.3 - Path Independence, Conservative Fields, and Potential Functions - Exercises - Page 850: 16

$-\pi$

We know that the line equation is defined as: $r(t)=r_0+kt$ Thus, $r_0+kt=(0,0,0)+t \lt 3,3,1 \gt=\lt 3t, 3t, t \gt$ So, $x=3t \implies dx= 3 dt$; $y=3 t \implies dy=3 dt$ and $z=2t \implies dz=2dt$ We need to substitute all the values in the given integral. $\int_{0,0,0}^{3,3,1}18tdt-27t^2dt+\dfrac{-4}{1+t^2} dt=-\pi$