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Sketch a circle tangent to the x-axis and compute its area from 0 to π

by RoRi

Define a circle tangent to the x-axis by:

    \[ f(\theta) = 4 \sin \theta, \qquad 0 \leq \theta \leq \pi. \]

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_0^{\pi} 16 \sin^2 \theta \, d \theta &= -4 \int_0^{\pi} (1 - 2 \sin^2 \theta - 1) \, d \theta \\ &= 4 \int_0^{\pi} d \theta - 4 \int_0^{\pi} \cos (2 \theta) \, d \theta \\ &= 4 \pi - 2 \int_0^{2 \pi} \cos \theta \, d \theta \\ &= 4 \pi. \end{align*}

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