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

User Nitarshan
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1 Answer

3 votes
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
User Ivan Besarab
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