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The problem involves setting up and evaluating the integral in spherical coordinates to find the volume of the solid enclosed by the cone \( z = \sqrt{x^2 + y^2} \) and bounded above by the sphere \( x^2 + y^2 + z^2 = 4 \).

### Steps to Solve:

1. **Convert the Equations to Spherical Coordinates:**
   - The cone equation \( z = \sqrt{x^2 + y^2} \) converts to \( \rho\cos\phi = \rho\sin\phi \), simplifying to \( \tan\phi = 1 \) or \( \phi = \pi/4 \).
   - The sphere equation \( x^2 + y^2 + z^2 = 4 \) converts to \( \rho^2 = 4 \) or \( \rho = 2 \).

2. **Set Up the Integral:**
   In spherical coordinates, the volume element is \( dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \). Thus, the integral becomes:
   \[
   V = \int_{0}^{2\pi} \int_{0}^{\pi/4} \int_{0}^{2} \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta
   \]

3. **Evaluate the Integral:**
   - Integrate \( \rho^2 \) with respect to \( \rho \) from 0 to 2.
   - Then integrate \( \sin\phi \) with respect to \( \phi \) from 0 to \( \pi/4 \).
   - Finally, integrate with respect to \( \theta \) from 0 to \( 2\pi \).

Let's perform these integrations.

### Performing Integration:The volume of the solid enclosed by the cone \( z = \sqrt{x^2 + y^2} \) and bounded above by the sphere \( x^2 + y^2 + z^2 = 4 \) is \( \frac{8}{3} \pi (1 - \sqrt{2}) \).

Would you like to explore this solution further or do you have any questions on this topic?

Here are some related questions you might consider:
1. How does the choice of limits impact the integral's value in spherical coordinates?
2. What would be the volume if the bounding sphere had a different radius?
3. How does the volume change if the cone's angle is different from \( \pi/4 \)?
4. Can you solve the same problem using cylindrical coordinates?
5. What are the applications of integrating in spherical coordinates in physics and engineering?

Tip: When integrating in spherical coordinates, visualize the solid to accurately set the limits for \( \rho \), \( \phi \), and \( \theta \).

Set up and evaluate the integral in spherical coordinates to find the volume of the solid bounded below by the cone z = sqrt(x^2 + y^2) and bounded above by the sphere x^2 + y^2 + z^2 = 4.

This problem involves finding the volume of a solid enclosed by a cone and a sphere using spherical coordinates. It covers concepts like volume integration in three dimensions, converting Cartesian equations to spherical coordinates, and applying multivariable calculus.

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