Mathegoras

Coordinate geometry › Polar and parametric curves › Areas in polar coordinates

Area under a rotating rectangle's path

High school · ●●●○○ 50 pts

A 2×32 \times 3 rectangle has vertices (0,0)(0,0), (2,0)(2,0), (0,3)(0,3), (2,3)(2,3). Rotate it 90∘90^{\circ} clockwise about (2,0)(2,0), then 90∘90^{\circ} clockwise about (5,0)(5,0), then 90∘90^{\circ} clockwise about (7,0)(7,0), and finally 90∘90^{\circ} clockwise about (10,0)(10,0), so that the side originally on the xx-axis is back on the xx-axis. A point of the rectangle initially at (1,1)(1,1) traces a curve during these rotations. Find the area of the region below this curve and above the xx-axis. Round to two decimal places.

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area of composite regionsquarter circlearea under a curve