Here is a simple problem which I would occasionally assign to my precalculus students and to my calculus students. The precalculus students always found a simpler answer. Sometimes it is possible to know too much. :)
Construct a simple proof that the area of the shaded region of the circle is $$ \frac{1}{2}\pi r^2+2ab $$
Caution! Mousing over the yellow region will reveal the answer.
Bonus: For those who got the answer or who revealed the answer, what does the dashed line represent? What is its equation?
As a variation on the same symmetry clues used in the posted spoiler, the areas of the white and shaded parts are, respectively:
$$ \begin{cases} \begin{align} S_W &= S' + 2 S'' + S''' \\ S_B &= S'+2 S''+S''' + S \end{align} \end{cases} $$
Therefore $S_B-S_W=S=4ab$ and since $S_B+S_W=\pi r^2$ the result immediately follows.