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

$-16$

We know that the line equation is defined as: $r(t)=r_0+kt$ Thus, $r_0+kt=(0,0,0)+t \lt 1,2,3 \gt=\lt t, 2t, -3t \gt$ So, $x=t \implies dx= dt$; $y=2t \implies dy=2dt$ and $z=3t \implies dz=3dt$ We need to substitute all the values in the given integral. $\int_{0,0,0}^{1,2,3}(4t^2)dt+2(t-9t^2)dt-(2)(2t)(3t)3 dt=-16$