Are the areas of the same size? The colored areas, are they of the same size?

Regards
 A: I'd look at whether their vertical cross-sections are of the same size.  If you draw a vertical line intersecting both yellow regions, ask whether its intersection with one of those regions is longer than its intersection with the other one.
A: Rotating the two 180 degrees and then translating the smaller circle down and making the black horizontal lines the same, you get two integrals:
$$\int_{0}^t (r_i-\sqrt{r_i^2-x^2}\ ) \ dx$$
For $i=1,2$ and fixed $t$. This is not gonna be equal for two different $r_i$.
Basically, the two circles never "cross" - they touch at a tangent at the origin - so one of these functions is always smaller than the other, so the integrals will always be different except when $t=0$.
Geometric statement: If two distinct circles are both tangent to line $\mathcal l$ at point $P$, then they cannot have another common point.
Proof: The centers of the two circles must be on the perpendicular to $\mathcal l$ at $P$. If there was another point $Q$ in common between the two circles, then the centers of both circles would also be on the perpendicular bisector of $PQ$. This would mean the circles share a center, and share a point, so they must be the same circle.
A: 1) Take two squares $A$ and $B$ and put a circle as big as possible inside each of them. Denote the circle inside $A$ with $C$ and the circle inside $B$ with $D$.
2) Calculate the areas of $A$, $B$, $C$ and $D$. Calculate $x = \frac{A - C}{4}$ and $y = \frac{A - D}{4}$ to get the area between the corner of a square and the circle in it.
3) The yellow areas in your figures are proportional to $x$ and $y$
4) If $x = y$ for every $A$ and $B$ so are the yellow areas are also the same.
It's just a layman shouting here, so correct me if I am wrong.
A: Obviously not. The lower region is strictly taller than the upper region everywhere.
