### Instruction: Two circles of radius 2 are centered at $(2,0)$ and at $(0,2)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$.### Response: I want to find the area of the shaded region in this diagram, which is the intersection of the interiors of the two circles.
I notice that the shaded region consists of two congruent segments, each of which is bounded by a chord and an arc of a circle.
I also notice that the chord and the arc subtend a right angle at the center of the circle, so the arc length is one-fourth of the circumference of the circle, and the area of the segment is one-fourth of the area of the circle minus the area of the right triangle formed by the chord and the radii.
Therefore, the area of one segment is $\frac{1}{4}\pi r^2 - \frac{1}{2}r^2$, where $r$ is the radius of the circle.
In this case, $r=2$, so the area of one segment is $\frac{1}{4}\pi (2)^2 - \frac{1}{2}(2)^2 = \pi - 2$.