Question

Evaluate the line integral ∫cx4zds, where c is the line segment from (0,6,5) to (8,4,1).

Answer 1

Parameterize the path $\mathcal C$ by

$\mathbf r(t)=\langle0,6,5\rangle(1-t)+\langle8,4,1\rangle t=\langle8t,6-2t,5-4t\rangle$

with $0\le t\le1$. Then

$\displaystyle\int_{\mathcal C}x^4z\,\mathrm dS=\int_{t=0}^{t=1}(8t)^4(5-4t)\sqrt{8^2+(-2)^2+(-4)^2}\,\mathrm dt=8192\sqrt{21}-16384\sqrt{\frac73}$