You have taken a sledgehammer to a peanut. It is a great credit to you that, with so many steps in which to make a mistake, you got the right answer.
My solution was worked out on one side of an index card, and involves nothing more complicated than arctan. Here is how it goes.
Let the line going through the center of both circles be our x axis, with the center of the larger circle at the origin. The center of the smaller circle is at (s, 0), where s is the square root of 2 over 2. Now, find the intersection of the circles, (x,y), using the formulae for the circles:
Solving, one gets, x equals the square root of 25/32, and y is the square root of 7/32.
Now find the angles for the sectors of each circle, a and b, as the arctans of y/x and y/(x-s), respectively. The areas of the sectors are a/2 and b/8. There is a triangle connecting the centers of the circles and (x, y). It’s area, is the distance between the centers, s, the height, y, divided by 2. So, half of the area we seek is:
Which, of course, is half of your “Bosome” number.
Actually, you even made your calculus approach much harder than necessary. Try it again with my x, y axes switched, to get the area between the two circles from the difference of the two integrals between minus and plus square of 7/32.