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Quarter intersection Problem (Posted on 2002-10-30) Difficulty: 5 of 5
Below is an image of a square, 7 cm to a side, with four quarter-circles inscribed into it as showm.

Find the area of the shaded region in the center.

See The Solution Submitted by Dulanjana    
Rating: 4.0667 (15 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Solution (can somebody check my math?) | Comment 7 of 9 |
(In reply to Solution (can somebody check my math?) by friedlinguini)

The solution does come out to the same value by using calculus. Using the large square again as a unit square, we can later multiply by 49. Dividing the large square into quarters, the central shaded area is also quartered. Taking the upper right part of this, related to the origin in the lower left corner of the original large square, we need the integral of √(1-x²)-.5 from .5 to √3/2. The integral is -.5x + .5x√(1-x²) + .5 arcsin(x). When this is evaluated from .5 to √3/2 the result is 0.07878669, which, multiplied by 4 and then by 49, is 15.44.
  Posted by Charlie on 2003-02-10 10:05:33

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