VC5.6

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If R is a plane region bounded by a simple closed curve C,then by Green's theorm in plane,we have \oint _{C}(\psi dx+\phi dy)=\iint _{R}[{\frac  {\partial \phi }{\partial x}}-{\frac  {\partial \psi }{\partial y}}]dxdy\, --(1)

putting \psi =-y,\phi =x\, in (1),we have

\oint _{C}(xdy-ydx)=\iint _{R}[{\frac  {\partial }{\partial x}}(x)-{\frac  {\partial }{\partial y}}(-y)]dxdy=2\iint _{R}dxdy=2A\, where A is the area of plane region R bounded by C.

Therefore,A={\frac  {1}{2}}\oint _{C}(xdy-ydx)\,

The parametric equations of the ellipse {\frac  {x^{2}}{a^{2}}}+{\frac  {y^{2}}{b^{2}}}=1\, are x=a\cos \theta ,y=b\sin \theta ,0\leq \theta \leq 2\pi \, -(2)

The area of the ellipse={\frac  {1}{2}}\oint _{C}(xdy-ydx)={\frac  {1}{2}}\int _{{\theta =0}}^{{2\pi }}[x{\frac  {\partial y}{\partial \theta }}-y{\frac  {\partial x}{\partial \theta }}]d\theta \,

={\frac  {1}{2}}\int _{{0}}^{{2\pi }}[(a\cos \theta )(b\cos \theta )-(b\sin \theta )(-a\sin \theta )]d\theta \,

={\frac  {1}{2}}\int _{{0}}^{{2\pi }}ab(\cos ^{2}\theta +\sin ^{2}\theta )d\theta \,

={\frac  {1}{2}}ab\int _{{0}}^{{2\pi }}d\theta =\pi ab\,

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