Question

Evaluate the integral by finding parametric equations for the line segment from ​(4​,4​,2​) to ​(5​,8​,negative 4​) and evaluating the line integral of fequalsyiplusxjplus3k along the segment.

Answer 1

Parameterize the line segment (call it $C$) by

$\vec r(t)=(1-t)(4,4,2)+t(5,8,-4)=(4+t,4+4t,2-6t)$

with $0\le t\le1$. Then the line integral of $\vec f(x,y)=y\,\vec\imath+x\,\vec\jmath+3\,\vec k$ is

$\displaystyle\int_C\vec f\cdot\mathrm d\vec r=\int_0^1(4+4t,4+t,3)\cdot(1,4,-6)\,\mathrm dt=\int_0^1(2+8t)\,\mathrm dt=6$