Using polar coordinates to find area of a circle

Question

Since the area of a polar curve is defined as:

$$ \int_a^b \frac 12 r^2 d\theta $$

and since $r$ is constant, independent of $\theta$, can this be re-written as?

$$ \frac 12 r^2 \int_a^b d\theta $$

Answer 1

If $r$ is actually independent of $\theta$, then yes. But then the curve is an arc of a circle centered at the origin, the area you are finding is that of a sector of a circle centered at the origin, and the whole thing is simple and not interesting.

In the general case, $r$ is not independent of $\theta$, so your manipulation is not valid. You usually need to do it the long way, substituting the formula for $r$ and finding the integral.

Answer 2

Circle centered at the pole and radius $a$, $r(\theta)=a$. Area is obtained by $$Area=\dfrac{a^2}{2}\int_0^{2\pi}d\theta=\pi{a^2}$$