# 06 Area Within the Curve r^2 = 16 cos θ

**Example 6**

What is the area within the curve r^{2} = 16 cos θ?

**Solution**

$r^2 = 16 \cos \theta$

θ |
0° | ±30° | ±60° | ±90° | > 90° |

r |
±4 | ±3.72 | ±2.83 | 0 | imaginary |

The values in the table show that the graph is symmetrical to the origin and θ ranges from -90° to 90°.

$A = {\displaystyle \frac{1}{2}{\int_{\theta_1}}^{\theta_2}} r^2 \, d\theta$

$A = 4\left[ {\displaystyle \frac{1}{2}{\int_0}^{\,90^\circ}} 16 \cos \theta \, d\theta \right]$

$A = 32 \Big[ \sin \theta \Big]_0^{90^\circ}$

$A = 32 \Big[ \sin 90^\circ - \sin 0^\circ \Big]$

$A = 32 \, \text{ units}^2$ *answer*

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