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The problem asks to set up the triple integral in spherical coordinates to calculate the volume of the solid \( E \), which is inside the surface \( S_1 \), outside \( S_2 \), and limited below by \( S_3 \).

Let's analyze the surfaces involved:

1. **\( S_1: x^2 + y^2 + z^2 = 25 \)**  
   This is a sphere centered at the origin with radius 5. In spherical coordinates, this becomes:
   \[
   \rho = 5
   \]

2. **\( S_2: z = \frac{\sqrt{x^2 + y^2}}{3} \)**  
   This represents a cone with an opening angle. In spherical coordinates, the equation of a cone is:
   \[
   \tan\theta = 1/3 \quad \Rightarrow \theta = \arctan(1/3)
   \]

3. **\( S_3: z = 0 \)**  
   This is the \( xy \)-plane, and in spherical coordinates, this corresponds to:
   \[
   \theta = \frac{\pi}{2}
   \]

### Setting up the triple integral:

In spherical coordinates, the volume element \( dV \) is given by:
\[
dV = \rho^2 \sin\theta \, d\rho \, d\theta \, d\phi
\]
where:
- \( \rho \) is the radial distance,
- \( \theta \) is the polar angle (measured from the positive \( z \)-axis),
- \( \phi \) is the azimuthal angle (measured from the positive \( x \)-axis in the \( xy \)-plane).

The limits of integration for the volume of solid \( E \) are:

- **For \( \rho \)**: The radial distance \( \rho \) goes from 0 to 5 (since the solid is bounded by the sphere \( S_1 \)).
  \[
  0 \leq \rho \leq 5
  \]
  
- **For \( \theta \)**: The polar angle \( \theta \) is limited by the cone \( S_2 \) and the plane \( S_3 \). From the cone equation, \( \theta = \arctan(1/3) \), so:
  \[
  \arctan(1/3) \leq \theta \leq \frac{\pi}{2}
  \]
  
- **For \( \phi \)**: Since the solid is symmetric around the \( z \)-axis, \( \phi \) ranges from 0 to \( 2\pi \).
  \[
  0 \leq \phi \leq 2\pi
  \]

### The triple integral:

The triple integral in spherical coordinates to compute the volume of the solid \( E \) is:

\[
V = \int_0^{2\pi} \int_{\arctan(1/3)}^{\frac{\pi}{2}} \int_0^5 \rho^2 \sin\theta \, d\rho \, d\theta \, d\phi
\]

This setup will allow you to calculate the volume of the solid.

Would you like to solve the integral or need further clarifications?

### Related Questions:
1. What is the meaning of the radial coordinate \( \rho \) in spherical coordinates?
2. How does the cone equation translate into spherical coordinates?
3. What are the properties of the volume element \( dV \) in spherical coordinates?
4. How would the limits of integration change if the radius of the sphere \( S_1 \) was different?
5. How could this problem be approached using cylindrical coordinates?

### Tip:
In spherical coordinates, always pay close attention to how angular limits are derived from the geometry of the problem, especially for cones and planes.

Sea el sólido E interior a S1, exterior a S2 y limitado inferiormente por S3. S1: x^2 + y^2 + z^2 = 25, S2: z = sqrt(x^2 + y^2) / 3, S3: z = 0. Plantee la integral triple en coordenadas esféricas que permite calcular el volumen del sólido E.

Learn how to set up a triple integral in spherical coordinates to calculate the volume of a solid defined by the surfaces S1, S2, and S3. This problem includes the conversion of equations into spherical coordinates and the integration limits.

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