Integration or Geometry:
a Problem for College Students
The Ariel University Center in Samaria (Israel) runs an online mathematical olympiad for college students. Last year there were several beautiful problems, one of which might appear harder for college students than for younger mathematicians:
The point P = P(a, b) is located in the first quadrant. Consider a circle centered at the point P with radius greater than √a² + b², and denote the area of the part of this circle located in the i-th quadrant (i = 1, 2, 3, 4) by Si . Find S1 - S2 + S3 - S4.
For a solution, scroll down.
For a solution, draw two extra lines: x = 2a and y = 2b. Together with the axes, these lines split the circle area into 9 pieces which (because of the symmetry) cancel out in S1 - S2 + S3 - S4, except for the central rectangle of area (2a)(2b) = 4ab.
An interactive illsutration is available elsewhere.
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