29.29. Calculate the volume $V$ of a sphere of radius $R$ using a definite integral.
29.29. Answer: $\frac{4}{3} \pi R^{3}$. A plane at a distance $x$ from the center of a sphere cuts a circle of radius $\sqrt{R^{2}-x^{2}}$ on the sphere. The area of this circle is $\pi\left(R^{2}-x^{2}\right)$, therefore

$$
V=\int_{-R}^{R} \pi\left(R^{2}-x^{2}\right) d x=2 \pi R^{3}-\left.\pi \frac{x^{3}}{3}\right|_{-R} ^{R}=2 \pi R^{3}-\frac{2 \pi R^{3}}{3}=\frac{4 \pi R^{3}}{3}
$$