Q7. The solid E lies between x^2+y^2+z^2=1 and x^2+y^2+z^2=4. The density function p(x,y,z)=z^2 (kg/m^3). Find the total mass of E (kg).

To find the total mass of the solid \( E \) bounded by the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 4 \) with the density function \( \rho(x,y,z) = z^2 \), we will use spherical coordinates to set up and evaluate the integral.

In spherical coordinates, the relationships are:
\[ x = \rho \sin \phi \cos \theta \]
\[ y = \rho \sin \phi \sin \theta \]
\[ z = \rho \cos \phi \]
where:
- \( \rho \) is the radial distance,
- \( \phi \) is the polar angle (0 to \( \pi \)),
- \( \theta \) is the azimuthal angle (0 to \( 2\pi \)).

The Jacobian determinant for the transformation to spherical coordinates is \( \rho^2 \sin \phi \).

The density function in spherical coordinates becomes:
\[ \rho(x,y,z) = z^2 = (\rho \cos \phi)^2 = \rho^2 \cos^2 \phi \]

The volume element in spherical coordinates is \( dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \).

Therefore, the mass \( M \) is given by:
\[ M = \int \int \int_E \rho(x,y,z) \, dV \]
\[ M = \int_0^{2\pi} \int_0^{\pi} \int_1^2 \rho^2 \cos^2 \phi \cdot \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \]

Combining and simplifying the integrand:
\[ M = \int_0^{2\pi} \int_0^{\pi} \int_1^2 \rho^4 \cos^2 \phi \sin \phi \, d\rho \, d\phi \, d\theta \]

We can separate the integrals because the limits are independent:
\[ M = \left( \int_0^{2\pi} d\theta \right) \left( \int_0^{\pi} \cos^2 \phi \sin \phi \, d\phi \right) \left( \int_1^2 \rho^4 \, d\rho \right) \]

Evaluating each integral:

1. The \(\theta\) integral:
\[ \int_0^{2\pi} d\theta = 2\pi \]

2. The \(\phi\) integral:
\[ \int_0^{\pi} \cos^2 \phi \sin \phi \, d\phi \]
Let \( u = \cos \phi \), hence \( du = -\sin \phi \, d\phi \):
\[ \int_0^{\pi} \cos^2 \phi \sin \phi \, d\phi = \int_1^{-1} u^2 (-du) = \int_{-1}^1 u^2 \, du \]
\[ \int_{-1}^1 u^2 \, du = 2 \int_0^1 u^2 \, du = 2 \left[ \frac{u^3}{3} \right]_0^1 = 2 \left( \frac{1^3}{3} - \frac{0^3}{3} \right) = \frac{2}{3} \]

3. The \(\rho\) integral:
\[ \int_1^2 \rho^4 \, d\rho = \left[ \frac{\rho^5}{5} \right]_1^2 = \frac{2^5}{5} - \frac{1^5}{5} = \frac{32}{5} - \frac{1}{5} = \frac{31}{5} \]

Multiplying all parts together:
\[ M = 2\pi \cdot \frac{2}{3} \cdot \frac{31}{5} = \frac{4\pi}{3} \cdot \frac{31}{5} = \frac{124\pi}{15} \]

Therefore, the total mass of the solid \( E \) is:
\[ \boxed{\frac{124\pi}{15} \text{ kg}} \]

Do you have any questions or need further details?

Here are 8 related questions for further exploration:
1. How do spherical coordinates simplify the integration process for spherical regions?
2. What is the volume of the solid E between the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 4 \)?
3. How would the calculation change if the density function were different, such as \( \rho(x,y,z) = z \)?
4. What is the significance of the Jacobian determinant in coordinate transformations?
5. Can this method be extended to find the mass of solids bounded by other types of surfaces, such as ellipsoids?
6. How would you approach the problem if the density function were \( \rho(x,y,z) = x^2 + y^2 \)?
7. How can we verify the correctness of our mass calculation?
8. What are some practical applications of finding the mass of a solid with a given density function?

**Tip:** When solving triple integrals, always check if transforming to spherical or cylindrical coordinates simplifies the limits and integrand.

Learn how to calculate the total mass of a solid E bounded by spheres with a given density function using triple integrals and spherical coordinates. Detailed step-by-step solution provided.

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