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 $$ \frac{1}{2}\pi r^2+2ab $$

Rectangled circle

Caution! Mousing over the yellow region will reveal the answer.

Rectangled Circle "Proof"

Bonus: For those who got the answer or who revealed the answer, what does the dashed line represent? What is its equation?

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    $\begingroup$ That's a neat way to showcase the symmetry. As for the dashed line, my guess would be $\,a\,b=const\,$ i.e. the locus of points which give the same shaded area. $\endgroup$ – dxiv Dec 16 '16 at 19:14
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    $\begingroup$ @dxiv That's correct. $\endgroup$ – John Wayland Bales Dec 16 '16 at 19:17
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    $\begingroup$ Too much knowledge can make you think like a robot after a while if you're not careful. In the spider in a cuboid room, I tried to minimize the distance function by setting its derivative equal to zero. More thinking required. mathblog.dk/project-euler-86-shortest-path-cuboid $\endgroup$ – John Joy Dec 16 '16 at 20:13


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.


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