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\|(\mathrm d\mathbf r(t))/(\mathrm dt)\right\|\,\mathrm dt`

`=\displaystyle\int_(t=0)^(t=\pi)(3\cos t)^2(3\sin t)√((-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*3^4\int_(u=0)^(u=1)u^2\,\mathrm du`

`=54`