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Problem 244
244
In the diagram, AYB is a semicircle with AB as diameter and OAXB is a sector of a circle with centre O and radius r. Angle AOB = 2θ radians. Find an expression, in terms of r and θ, for the area of the shaded region.
Solution
The radius of the semicircle is \(\frac{1}{2} AB = r \sin \theta\).
The area of the semicircle is \(\frac{1}{2} \pi r^2 \sin^2 \theta = A_1\).
The shaded area is the area of the semicircle minus the area of the segment.