Lately, I've been trying to come up with tricks to solve integrals quickly.
So let's say I have $$\int_{0}^{2\pi} \cos^2 \theta d\theta$$
Now if I were to look at this integral in polar coordiantes, I get
$$\frac{1}{2}\int_{0}^{2\pi} \cos^2 \theta d\theta$$
The integrand is a circle in polar coordinates $r = 2(1/2)\cos\theta$ with radius $1/2$
So the integral $$\frac{1}{2}\int_{0}^{2\pi} \cos^2 \theta d\theta = \frac{\pi}{2}$$
But this doesn't make sense to me because the area should be $\pi (1/2)^2 = \pi/4$
What's going on? I am trying to extend this idea to $$\int_{0}^{2\pi} \sin^2 \theta d\theta$$ and linear combinations of sine and cosines squared