Exercise 1203Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Area using Green's theorem

#area#Green#closed curve

Statement

Use Green's theorem to compute the area of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .

Hint

Area $=\frac12\oint_C(x\,dy-y\,dx)$ .

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Parametrization. $\gamma(t)=(a\cos t,b\sin t)$ , $t\in[0,2\pi]$ .

Compute $x\,dy-y\,dx$ . $x\,dy=ab\cos^2t\,dt$ , $y\,dx=ab\sin^2t\,dt$ .

Integral. $\frac12\int_0^{2\pi}ab\,dt=\pi ab$ .

Conclusion. Area $=\pi ab$ .