To evaluate the integral

$\iiint_E (x^2 + y^2) \, dV$

where $E$ lies between the spheres $x^2 + y^2 + z^2 = 1$ and $x^2 + y^2 + z^2 = 16$, we can use spherical coordinates.

Step 1: Setting up the spherical coordinates

In spherical coordinates:

• $x = \rho \sin \theta \cos \phi$
• $y = \rho \sin \theta \sin \phi$
• $z = \rho \cos \theta$
• The volume element $dV$ in spherical coordinates is given by $dV = \rho^2 \sin \theta \, d\rho \, d\theta \, d\phi$.

In these coordinates, $x^2 + y^2 = \rho^2 \sin^2 \theta$ and the equation for the spheres becomes:

• Inner sphere: $\rho = 1$
• Outer sphere: $\rho = 4$

Thus, $\rho$ ranges from 1 to 4, $\theta$ from 0 to $\pi$, and $\phi$ from 0 to $2\pi$.

Step 2: Set up the integral

The integral now becomes

$\iiint_E (x^2 + y^2) \, dV = \int_{0}^{2\pi} \int_{0}^{\pi} \int_{1}^{4} \rho^2 \sin^2 \theta \cdot \rho^2 \sin \theta \, d\rho \, d\theta \, d\phi$

Simplify the integrand:

$= \int_{0}^{2\pi} \int_{0}^{\pi} \int_{1}^{4} \rho^4 \sin^3 \theta \, d\rho \, d\theta \, d\phi$

Step 3: Separate the integrals

Since the integrand is separable in terms of $\rho$, $\theta$, and $\phi$, we can write:

$\int_{0}^{2\pi} \, d\phi \int_{0}^{\pi} \sin^3 \theta \, d\theta \int_{1}^{4} \rho^4 \, d\rho$

1. Evaluate the $\phi$-integral:

   $\int_{0}^{2\pi} \, d\phi = 2\pi$

2. Evaluate the $\rho$-integral:

   $\int_{1}^{4} \rho^4 \, d\rho = \left[ \frac{\rho^5}{5} \right]_{1}^{4} = \frac{4^5}{5} - \frac{1^5}{5} = \frac{1024}{5} - \frac{1}{5} = \frac{1023}{5}$

3. Evaluate the $\theta$-integral:

   $\int_{0}^{\pi} \sin^3 \theta \, d\theta$

To solve this, use the identity $\sin^3 \theta = \sin \theta (1 - \cos^2 \theta)$ and then apply integration techniques, or use a table of integrals. The result is:

$\int_{0}^{\pi} \sin^3 \theta \, d\theta = \frac{4}{3}$

Step 4: Combine the results

Putting it all together:

$\iiint_E (x^2 + y^2) \, dV = 2\pi \cdot \frac{4}{3} \cdot \frac{1023}{5} = \frac{8196\pi}{15}$

Thus, the evaluated integral is:

$\boxed{\frac{8196\pi}{15}}$

Would you like more details or have questions about any specific part of the solution?

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Here are some related questions:

1. How are spherical coordinates derived from Cartesian coordinates?
2. What applications require the use of spherical coordinates?
3. How would the limits change if the region were bounded by a cylinder instead of spheres?
4. Why does the volume element $dV = \rho^2 \sin \theta \, d\rho \, d\theta \, d\phi$ include $\rho^2$ and $\sin \theta$?
5. What other coordinate systems are commonly used in multivariable calculus, and how do they compare to spherical coordinates?

Tip: In spherical coordinates, always carefully analyze the bounds of $\rho$, $\theta$, and $\phi$, as they vary based on the region of integration.