Q2. Find the volume of the solid bounded by x=8-y^2-z^2 and x=y^2+z^2.

To find the volume of the solid bounded by the surfaces \( x = 8 - y^2 - z^2 \) and \( x = y^2 + z^2 \), we will use cylindrical coordinates. 

First, let's express the given equations in cylindrical coordinates:

\[
x = 8 - r^2
\]

\[
x = r^2
\]

where \( r^2 = y^2 + z^2 \).

In cylindrical coordinates, the volume integral is:

\[
V = \int \int \int dV = \int \int \int r \, dz \, dr \, d\theta
\]

The surfaces intersect where \( 8 - r^2 = r^2 \), so:

\[
8 = 2r^2 \implies r^2 = 4 \implies r = 2
\]

The bounds for \( r \) are from 0 to 2, for \( \theta \) from 0 to \( 2\pi \), and for \( z \) from the lower surface \( z = -\sqrt{8 - x} \) to the upper surface \( z = \sqrt{8 - x} \).

We need to integrate \( x = r^2 \) to \( x = 8 - r^2 \):

\[
V = \int_0^{2\pi} \int_0^2 \int_{r^2}^{8-r^2} r \, dx \, dr \, d\theta
\]

However, since we are using cylindrical coordinates and \( x \) is replaced by the equation for the height differential \( dz \):

\[
V = \int_0^{2\pi} \int_0^2 \int_{r^2}^{8-r^2} r \, dz \, dr \, d\theta
\]

We need to be cautious here; \( dz \) range is \( 2 \sqrt{8 - r^2} \):

\[
V = \int_0^{2\pi} \int_0^2 (8 - 2r^2) \, r \, dr \, d\theta
\]

Solving the integral:

1. Integrate with respect to \( r \):

\[
V = \int_0^{2\pi} \int_0^2 (8r - 2r^3) \, dr \, d\theta
\]

\[
= \int_0^{2\pi} \left[ 4r^2 - \frac{r^4}{2} \right]_0^2 \, d\theta
\]

\[
= \int_0^{2\pi} \left[ 4(2^2) - \frac{(2^4)}{2} \right] \, d\theta
\]

\[
= \int_0^{2\pi} \left[ 16 - 8 \right] \, d\theta
\]

\[
= \int_0^{2\pi} 8 \, d\theta
\]

2. Integrate with respect to \( \theta \):

\[
V = 8 \theta \bigg|_0^{2\pi} = 8 \times 2\pi = 16\pi
\]

Therefore, the volume of the solid is \( \boxed{16\pi} \).

Would you like further details or have any questions?

Here are 8 related questions for further exploration:

1. How would the volume change if the bounds for \( x \) were modified?
2. What is the geometric interpretation of the solid formed by these bounds?
3. How can this problem be solved using Cartesian coordinates?
4. What are some applications of cylindrical coordinates in physics?
5. How would the volume change if the functions were shifted?
6. How do you convert integrals from Cartesian to cylindrical coordinates?
7. What other shapes could be formed using similar bounded surfaces?
8. How does symmetry help in solving these types of integrals?

**Tip:** When converting integrals from Cartesian to cylindrical coordinates, always make sure to include the Jacobian determinant \( r \) to account for the area element in polar coordinates.

Learn how to calculate the volume of a solid bounded by the surfaces x=8-y^2-z^2 and x=y^2+z^2 using cylindrical coordinates and triple integrals. This problem explores advanced topics in calculus and solid geometry.

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