Chapter 6 - Applications of the Integral - 6.2 Setting Up Integrals: Volume, Density, Average Value - Exercises - Page 298: 58

$\frac{\pi R^{2}}{3} $

The average area of the circles whose radii vary from $0$ to $R$ is: $$\frac{1}{R-0} \int_{0}^{R}A(r)~dr$$ where $A(r)=\pi r^{2}$ is the area of the circle $$\frac{1}{R-0} \int_{0}^{R}\pi r^{2}~dr$$ $$\frac{\pi}{R-0} \int_{0}^{R} r^{2}~dr$$ $$\frac{\pi}{R} \int_{0}^{R} r^{2}~dr$$ $$\frac{\pi}{R} [\frac{r^{3}}{3}]_{0}^{R} $$ $$\frac{\pi}{R} (\frac{R^{3}}{3}-\frac{0^{3}}{3}) $$ $$\frac{\pi R^{2}}{3} $$