Mathematics

Mathematics

Problem

Consider the polar curves r=2cos⁡θr=2\cos\theta and r=2sin⁡θr=2\sin\theta. The region common to the two curves is a leaf-shaped area that lies in the first quadrant and is bounded by both graphs. (a) Identify all lines or axes with respect to which this bounded region is symmetric. Support each claim with a brief verbal justification based on the equations of the curves. (b) Explain how any symmetries you identified in part (a) affect the choice of limits of integration when setting up a definite integral for the area of the region. (c) State a single interval of length π4\dfrac{\pi}{4} or less that is sufficient for computing the entire area, and indicate the numerical factor by which the resulting integral should be multiplied to obtain the total area.
Intuitive explanation

Both curves are circles in disguise—r=2cos⁡θr=2\cos\theta is centered at (1,0)(1,0) and r=2sin⁡θr=2\sin\theta at (0,1)(0,1)—and they intersect where their rr values are equal. The leaf-shaped region is bounded by one curve on the left and the other on the right, so you'll integrate the difference of their squared radii using the polar area formula A=12∫r2 dθA=\frac{1}{2}\int r^2\,d\theta, carefully identifying which curve dominates in which angular interval.

Intuitive explanation

Both curves are circles in disguise—r=2cos⁡θr=2\cos\theta is centered at (1,0)(1,0) and r=2sin⁡θr=2\sin\theta at (0,1)(0,1)—and they intersect where their rr values are equal. The leaf-shaped region is bounded by one curve on the left and the other on the right, so you'll integrate the difference of their squared radii using the polar area formula A=12∫r2 dθA=\frac{1}{2}\int r^2\,d\theta, carefully identifying which curve dominates in which angular interval.

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