Question

Evaluate the line integral for x^2yds where c is the top hal fo the circle x62 _y^2 = 9

Answer 1

Parameterize $C$ by

$\mathbf r(t)=\langle x(t),y(t)\rangle=\langle3\cos t,3\sin t\rangle$

where $0\le t\le\pi$. Then the line integral is

$\displaystyle\int_Cx^2y\,\mathrm dS=\int_{t=0}^{t=\pi}x(t)^2y(t)\left\|\frac{\mathrm d\mathbf r(t)}{\mathrm dt}\right\|\,\mathrm dt$
$=\displaystyle\int_{t=0}^{t=\pi}(3\cos t)^2(3\sin t)\sqrt{(-3\sin t)^2+(3\cos t)^2}\,\mathrm dt$
$=\displaystyle3^4\int_{t=0}^{t=\pi}\cos^2t\sin t\,\mathrm dt$

Take $u=\cos t$, then

$=\displaystyle-3^4\int_{u=1}^{u=-1}u^2\,\mathrm du$
$=\displaystyle3^4\int_{u=-1}^{u=1}u^2\,\mathrm du$
$=\displaystyle2\times3^4\int_{u=0}^{u=1}u^2\,\mathrm du$
$=54$