11.091. On the segment $A B$ as a diameter, a semicircle is constructed with the center at point $O$, and on the segments $O A$ and $O B$, two semicircles are constructed, located in the same half-plane with boundary $A B$ as the first one. Find the surface area and volume of the figure formed by rotating around $A B$ the figure bounded by these three semicircles, if $A B=20$ cm.
## Solution.

The volume of the desired figure is: $V=V_{\boldsymbol{w}_{1}}-2 V_{\boldsymbol{w}_{2}}$, where $V_{\boldsymbol{w}_{1}}=\frac{4}{3} \pi R_{1}^{3}$ (Fig. 11.79). Since $AB=20 \text{ cm}$, then $R_{1}=10 \text{ cm}$. Thus, we get $V_{\boldsymbol{w}_{1}}=\frac{4000}{3} \pi$. Since $R_{1}=10 \text{ cm}$, then $R_{2}=5 \text{ cm}$. Therefore, $V_{\text {w }_{2}}=\frac{4}{3} \pi R_{2}^{3}=\frac{500}{3} \pi$. Then $V=\frac{4000}{3} \pi-\frac{1000}{3} \pi=1000 \pi \text{ cm}^{3}$. The surface area of the desired figure is: $S=S_{\text{w}_{1}}+2 S_{\text{w}_{2}}$. The surface area of the larger spherical surface is: $S_{\text{w}_{1}}=4 \pi R_{1}^{2}=400 \pi$. The surface area of the smaller spherical surface is: $S_{\text{w}_{2}}=4 \pi R_{2}^{2}=100 \pi$. Then

$$
S=400 \pi+200 \pi=600 \pi \text{ cm}^{2}
$$

Answer: $600 \pi \text{ cm}^{2} ; 1000 \pi \text{ cm}^{3}$.