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?
$$ \frac 12 r^2 \int_a^b d\theta $$
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$\begingroup$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.
$\endgroup$ $\begingroup$Circle centered at the pole and radius $a$, $r(\theta)=a$. Area is obtained by $$Area=\dfrac{a^2}{2}\int_0^{2\pi}d\theta=\pi{a^2}$$
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