how is the line calculated? this is too much of a sketch to do anything concrete with
How to calculate the surface area of the red part?
I know for a fact it can be done. There is a square, a circle and another line that you can construct using a compass.
I don't know how exactly, though. So here would be my actions:
1. Call friend who is getting a phd in mathematics.
2. Describe the problem.
3. Calculate using his instructions.
4. Profit.
ok
I think it's easiest in this coordinate system
the smaller circle has centre in (0,0) and radius 5
5² = y² + x²
or
y = ±√(5² - x²)
the larger circle (well, just the arc) has centre in (0,-5√2) and radius 10
10² = (y + 5√2)² + x²
or
y = ±√(10² - x²) - 5√2
(the first is implicit formula and the second in explicit formula; in both cases only the +√ branch of the explicit formula, the one in upper semiplane, is relevant)
the circles intersect in points where x² is the same in both implicit formulas:
5² - y² = 10² - (y + 5√2)²
(y + 5√2)² - y² = 10² - 5²
y² + 10√2y + 50 - y² = 75
10√2y = 25
y = 5/2√2
calculating x of intersections:
5² = y² + x²
25 = 25/8 + x²
x² = 25(7/8)
x = ±5√7/2√2
now you want to integrate between those intersections (from -x to x) the difference between integrals of both circles' explicit formulas:
ʃ √(5² - x²) dx - ʃ √(10² - x²) - 5√2 dx
indefinite integral of √(r² - x²) you can find in integral tables;
(fuck this is getting ugly. you get the idea. if you can't solve it from here on, maybe I go come return and finish it tomorrow. good night bernd.group)
(protip: you can integrate only from 0 to x since shape is symmetric, total area is twice that)
