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I think the formula is
$$A = \frac 1 2 \int_{\alpha}^{\beta} (\text{outer})^2 - (\text{inner})^2 d\theta$$
where $\alpha, \beta$ are where they intersect in $[0, 2\pi]$.
This is what I got based on that
$$A = \frac 1 2 \int_{3\pi/4}^{5\pi/4} (1+\sin \theta)^2 - (1+\cos \theta)^2 d\theta$$
Is that right?
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