Using Green’s Theorem to Find Area

Use Green’s Theorem to find the area of the region enclosed by the ellipse \[ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\qquad a>0\quad b>0 \]

Solution We express the ellipse using the parametric equations \(x=a\;\cos t\), \(y=b\;\sin t\), \(0\leq t\leq 2\pi \). Then \(dx=-a\;\sin t\,dt\) and \(dy=b\;\cos t\,dt\). Now we use (1) to find the area. \[ \begin{eqnarray*} A& =&\dfrac{1}{2}\oint_{C}(-y\,dx+x\,dy)=\dfrac{1}{2}\int_{0}^{2\pi }[-b\sin t(-a\sin t\,dt)+a\cos t(b\cos t\,dt)] \\ & =&\dfrac{1}{2}\int_{0}^{2\pi }ab(\sin ^{2}t+\cos ^{2}t)\,dt=\dfrac{1}{2}(ab)\int_{0}^{2\pi }dt=\dfrac{1}{2}ab(2\pi )=\pi ab \end{eqnarray*} \]