Date modified: Mon 2023-09-18 . 09:01 AM
Date created: Sat 2023-10-07 . 09:02 PM
# A geometry challenge: Four intersecting quarter circles. Some 20+ years ago my friend Johnny (possibly his real name) came to me excitedly and told me the following area can only be found using integrals, the common area of four intersecting quarter circles of radius $1$ with centers at the four corners of a square $(0,0),(1,0),(0,1),(1,1)$: ![[1 teaching/smc-summer-2023-math-8/problems/week-3/---files/week-3-wednesday-problems 2023-07-05 16.47.53.excalidraw.svg]] %%[[1 teaching/smc-summer-2023-math-8/problems/week-3/---files/week-3-wednesday-problems 2023-07-05 16.47.53.excalidraw|🖋 Edit in Excalidraw]], and the [[smc-summer-2023-math-8/problems/week-3/---files/week-3-wednesday-problems 2023-07-05 16.47.53.excalidraw.dark.svg|dark exported image]]%% One can set up some integral involving expressions such as $\sqrt{1-x^{2}}$, as it would involve circular arcs. The integral set up will also require carefully finding the intersection points. At the time, I haven't learned how to deal with integrals like this (Johnny was a year ahead of me). But I was able to find the area just using geometry. **Find this area exactly** (leave any $\pi$ or square roots etc. in it, not just decimal approximation) with whatever method you choose, and show your work. (You can use geometry or integrate.)