Question

Use Theorem 9.13 to show that the area of the circle defined by the polar equation $r=a$is ${\mathrm{\pi a}}^2$.

Short answer

The area of the circle is${\mathrm{\pi a}}^2$

Step-by-step solution

Step 1. Given Information

The given function of the circle is$r=a$.

Step 2. Use the formula for area of a region bounded by a curve

• The region is bounded by the curve $r=a$.
• The region is a circle, so the boundary arcs will be $\theta =0,\theta =2\pi$.
• So, the area of the region will be calculated as follows:role="math" localid="1649917065658" $\begin{array}{rcl}\frac{1}{2}{\int }_0^{2\mathrm{\pi }}{a}^{2}d\theta & =& \frac{a^2}{2}{\int }_0^{2\mathrm{\pi }}d\theta \\ & =& \frac{a^2}{2}\times 2\mathrm{\pi }\\ & =& {\mathrm{\pi a}}^2\end{array}$
• Hence, the area of the curve is${\mathrm{\pi a}}^2$.