integral

**mahefo** · November 28th 2009, 02:26 PM

Help me calculate this integral
$\int_0^{2\pi} \sqrt{a^2\sin^2 x+ b^2 \cos^2 x}dx$
Thanks so much.

**Media_Man** · November 29th 2009, 10:41 AM

There is no known method for solving this integral explicitly. Your best bet is using a numerical method. Consider...

The length of an arc defined by the vector equation $\vec{x}(t)$ from t=A to t=B is $L=\int_A^B\sqrt{\left(\frac{dx}{dt}\right)^2+\left (\frac{dy}{dt}\right)^2}dt$

So the integral you have posed is equivalent to asking the circumference of the ellipse defined by $\vec{x}(t)=(-a\cos t,b\sin t)$ . Take the derivatives to see what I mean. As you probably know, the circumference of an ellipse cannot be given in closed form.

You can either use an infinite series or approximation, which can be found here: Ellipse - Wikipedia, the free encyclopedia

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