Question

Find f · dr c for the given f and

Answer 1

Parameterize the path $\mathcal C$ by

$\mathbf r(t)=x(t)\,\mathbf i+y(t)\,\mathbf j$

with

$x(t)=3\cos t$
$y(t)=3\sin t$

and $0\le t\le\pi$. Then

$\displaystyle\int_{\mathcal C}\mathbf f(x,y)\cdot\mathrm d\mathbf r=\int_{t=0}^{t=\pi}\mathbf f(x(t),y(t))\cdot\dfrac{\mathrm d\mathbf r}{\mathrm dt}\,\mathrm dt$
$=\displaystyle\int_0^\pi(9\cos^2t\,\mathbf i+9\sin^2t\,\mathbf j)\cdot(-3\sin t\,\mathbf i+3\cos t\,\mathbf j)\,\mathrm dt$
$=27\displaystyle\int_0^\pi(\cos t\sin^2t-\sin t\cos^2t)\,\mathrm dt=-18$