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Sketch a “cardioid” and compute its area from 0 to 2 π

by RoRi

Define a cardioid by:

    \[ f(\theta) = 1+\cos \theta, \qquad 0 \leq \theta \leq 2 \pi. \]

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

The sketch is as follows:

Rendered by QuickLaTeX.com

Then, we compute the area,

    \begin{align*} a(R) = \frac{1}{2} \int_0^{2 \pi} (1 + 2 \cos \theta + \cos^2 \theta) \, d \theta &= \frac{1}{2} \int_0^{2 \pi} d \theta + \int_0^{2 \pi} \cos \theta \, d \theta + \frac{1}{2} \int_0^{2 \pi} \cos^2 \theta \, d\theta \\ &= \pi + 0 + \frac{\pi}{2} \\ &= \frac{3 \pi}{2} \end{align*}

where we know \int_0^{2 \pi} \cos^2 \theta \, d \theta = \pi from this exercise (Section 2.11, Exercise #7).

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