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Sketch a circle tangent to the y-axis and compute its area from -π/2 to π/2

by RoRi

Define a circle tangent to the y-axis by:

    \[ f(\theta) = 2 \cos \theta, \qquad - \pi/2 \leq \theta \leq \pi/2. \]

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

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Then, we compute the area,

    \begin{align*} a(R) = \frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 4 \cos^2 \theta \, d \theta &= 2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (1 - \sin^2 \theta) \, d \theta \\ &= 2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} d \, \theta + \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (1 - 2 \sin^2 \theta - 1) \, d \theta \\ &= 2 \pi + \left( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos (2 \theta) \, d\theta \right) - \pi \\ &= \pi + \frac{1}{2} \int_{-\pi}^{\pi} \cos \theta \, d \theta \\ &= \frac{1}{2} \left( \sin \theta \biggr \rvert_{-\pi}^{\pi} \right) + \pi \\ &= \pi. \end{align*}

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