Involute of a circle by a sheep

Calculus Level 5

A sheep is tied in a silo of radius rr, by a rope long enough to reach the opposite end of the silo(see figure above). The total area available for the sheep to graze can be written as,


where pp and qq are co-prime positive integers. What is the value of p+qp+q?

Note: The involute (of a circle), is the locus of points traced out by a taut string.
I found this problem in a book, and i thought i would share it.


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