# How to calculate $\int_0^{2\pi} \sqrt{1 - \sin^2 \theta}\;\mathrm d\theta$

How to calculate:

$$\int_0^{2\pi} \sqrt{1 - \sin^2 \theta}\;\mathrm d\theta$$

-
You could use the Pythagorean identity to simplify your integrand... –  Guess who it is. May 17 '11 at 16:13
The identity $1 - \sin^2{\theta} = \cos^2{\theta}$ should prove useful =) –  Adrián Barquero May 17 '11 at 16:14
Additionally, you could reduce it to the integral $$2\int_{\pi/2}^{3\pi/2}\sqrt{1-\sin^2\theta}\mathrm d\theta$$ –  Guess who it is. May 17 '11 at 16:15
More elaborate hint: don't blindly use the Pythagorean identity. Plot the integrand and then apply the identity accordingly. –  Guess who it is. May 17 '11 at 16:20
@J.M.: Is the potential pitfall blindly using the trig identity, or thinking that $\sqrt{x^2}=x$ rather than $\sqrt{x^2}=|x|$? –  Isaac May 17 '11 at 22:22

Hint: note that $$\sqrt{1-\sin^2\theta}=\sqrt{\cos^2\theta}=|\cos\theta|.$$

-

Use the fact that $\cos^{2}\theta = 1-\sin^{2}\theta$ and the fact that integral of $\cos\theta$ is $\sin\theta$. Also $\sqrt{1-\sin^{2}\theta} = |\cos{\theta}|$. And note that $\cos\theta$ is positive in the first and the fourth quadrant.

-
The integral of $\cos{\theta}$ is $\sin{\theta}$ not $-\sin{\theta}$ :) –  Osama Gamal May 17 '11 at 16:33
@Osama: Sorry. I have rectified it now. Thanks for pointing out. –  user9413 May 17 '11 at 16:34

Use Wolfram Alpha! Plug in "integrate sqrt(1-sin^2(x))". Then press "show steps". You can enter the bounds by hand...

http://www.wolframalpha.com/input/?i=integrate+sqrt%281-sin%5E2%28x%29%29

-
Great tool, thanks for the tip –  bobobobo Jan 28 '12 at 17:23

\begin{align} \int_0^{2\pi} \sqrt{1 - \sin^2 \theta} d\theta &= \int_0^{2\pi} \sqrt{\cos^2 \theta} d\theta \\ &= \int_0^{2\pi} | \cos \theta | d\theta \\ &= 4 \int_0^{\frac{pi}{4}} \cos \theta d\theta \\ &= 4 \end{align}

-