Chapter 6: Applications of Definite Integrals - Section 6.1 - Volumes Using Cross-Sections - Exercises 6.1 - Page 323: 55

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

The area of the cross-section of the hemisphere is: $A=R^2\pi- h^2 \pi=\pi (R^2-h^2)$ We integrate the integral to calculate the volume as follows: $V= V_{cylinder}-V_{cone}$ or, $=\pi R^2- \dfrac{ \pi R^2}{3} (R)$ or, $=\dfrac{2 \pi R^3}{3}$