2006 SMT/General Problems/Problem 14

Solution

Noticing that the first equation, $x^2+y^2\le\pi^2$ is the equation of a disk of radius $\pi$ , and that the graph of $y = sin(x)$ is an odd function which will divide the circle into two regions; an upper and lower half of equal areas, the solution is half the area of the circle (The area of upper section the be specific as the question ask for $y$ which satisfy $y\ge sin(x)$ ). $\frac{\pi \cdot r^2}{2} = \frac{\pi \cdot (\pi)^2}{2} = \boxed{\frac{\pi^3}{2}}$

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2006 SMT/General Problems/Problem 14

Solution