Example 3. Find the volume of the body $\Omega$, bounded by the surfaces

$$
z=\sqrt{36-x^{2}-y^{2}}, \quad z=\sqrt{\frac{x^{2}+y^{2}}{3}}
$$
Solution.

1. Since $\Omega$ is the region bounded by the upper hemisphere and the upper half-cone, it is convenient to switch to spherical coordinates

$$
\left\{\begin{array}{l}
x=\varrho \cos \varphi \sin \theta \\
y=\varrho \sin \varphi \sin \theta \\
z=\varrho \cos \theta
\end{array}\right.
$$

In this case, $(\varrho, \theta, \varphi) \in \Omega^{\prime}$, and the desired volume is determined by the formula

$$
V=\iiint_{\Omega^{\prime}} \varrho^{2} \sin \theta d \varrho d \theta d \varphi
$$

Substitute $x$ with $\varrho \cos \varphi \sin \theta$, $y$ with $\varrho \sin \varphi \sin \theta$, and $z$ with $\varrho \cos \theta$ in the surface equations. After transformations, we get

$$
\varrho=6, \quad \operatorname{tg} \theta=\sqrt{3}
$$

The region $\Omega^{\prime}$ is bounded by these surfaces.

2. Define the region $\Omega^{\prime}$ using a system of inequalities

![](https://cdn.mathpix.com/cropped/2024_05_22_5878ce417e2ca061000bg-328.jpg?height=170&width=549&top_left_y=1429&top_left_x=350)

3. Calculate the volume using formula (3), reducing the triple integral to a repeated integral:

$$
V=\int_{0}^{2 \pi} d \phi \int_{0}^{\pi / 3} \sin \theta d \theta \int_{0}^{6} \varrho^{2} d \varrho=72 \pi
$$

Answer. $V=72 \pi$ cubic units.

Problem conditions. Find the volumes of the bodies bounded by the surfaces

1. $x=2 \sqrt{y}$,
$x=\sqrt{y}$,
$z=0$,
$z=1-y$.
2. $y=2 \sqrt{x}$,
$y=\sqrt{x}$,
$z=0$,
$z=1-x$.
3. $x^{2}+y^{2}=5$,
$y=2 \sqrt{x}$,
$z=0$,
$z=2 x$.
4. $x^{2}+y^{2}-2 x=0$,
$z=0$,
$z=4-y^{2}$.
5. $x^{2}+y^{2}-2 y=0$,
$z=0$,
$z=4-x^{2}$.
6. $z=\sqrt{4-x^{2}-y^{2}}$,
$z=\sqrt{3}\left(x^{2}+y^{2}\right)$.
7. $z=\sqrt{9-x^{2}-y^{2}}, \quad z=0$,
$x^{2}+y^{2}=1 \quad\left(x^{2}+y^{2} \leq 1\right)$.
8. $z=\sqrt{9-x^{2}-y^{2}}, \quad z=\sqrt{\frac{x^{2}+y^{2}}{3}}$.
9. $z=\sqrt{4-x^{2}-y^{2}}, \quad z=\sqrt{\frac{x^{2}+y^{2}}{24}}$.
10. $z=x^{2}+y^{2}$,
$z=\sqrt{x^{2}+y^{2}}$.

Answers. 1. $V=4 / 15$. 2. $V=4 / 15$. 3. $V=8 / 5$. 4. $V=15 \pi / 4$. 5. $V=15 \pi / 4$. 6. $V=2 \pi(8 / 3-5 \sqrt{3} / 4)$. 7. $V=2 \pi(9-16 \sqrt{2} / 3)$. 8. $V=9 \pi$. 9. $V=64 \pi / 5$. 10. $V=\pi / 6$.

### 12.13. Calculation of the Mass of a Body

Problem Statement. Find the mass of the body $\Omega$ with density $\mu(x, y, z)$, bounded by the given surfaces.

SOLUTION PLAN.

1. The mass of the body $\Omega$ with density $\mu(x, y, z)$ is determined by the formula

$$
m=\iiint_{\Omega} \mu(x, y, z) d x d y d z
$$

2. Define the region $\Omega$ using inequalities.
3. Calculate the triple integral, reducing it to a repeated integral, and write the answer, not forgetting the units of measurement.