A circle is centred at $(a,b)$ and has radius $d$. The value $k$ is such that the lines $y =k $ and $x=k$ both intersect the circle twice. Moreover, $a,b<k$, so that the point $(k,k)$ is inside the circle, but also above, and to the right of, the point $(a,b)$. This setup is illustrated below.
Given this information, what is the area of the shaded region?
I tried to do this geometrically, but couldn't work out the appropriate angles. Then I attempted it by using integrals, but this also became very complicated very quickly. It seems like there should be a neat way of seeing what this area is, but I can't seem to find it.