Solve ∫ E Z 2 D X D Y D Z \int_Ez^2dxdydz ∫ E ​ Z 2 D X D Y D Z Where E = { ( X , Y , Z ) : X 2 + Y 2 > A 2 , X 2 + Y 2 + Z 2 < 4 A 2 } E=\{(x,y,z):x^2+y^2>a^2,x^2+y^2+z^2<4a^2\} E = {( X , Y , Z ) : X 2 + Y 2 > A 2 , X 2 + Y 2 + Z 2 < 4 A 2 }

Introduction

In this article, we will delve into the world of real analysis and Lebesgue integrals, focusing on solving a specific triple integral. The given integral is $\int_Ez^2dxdydz$, where $E=\{(x,y,z):x^2+y^2>a^2,x^2+y^2+z^2<4a^2\}$. We will utilize the change of variables theorem to simplify the integral and arrive at a solution.

Understanding the Region of Integration

Before we proceed with the integration, let's analyze the region of integration, denoted by $E$. The region $E$ is defined as the set of points $(x,y,z)$ that satisfy the following conditions:

• $x^2+y^2>a^2$
• $x^2+y^2+z^2<4a^2$

The first condition represents a circle centered at the origin with radius $a$, while the second condition represents a sphere centered at the origin with radius $2a$. The region $E$ is essentially the intersection of these two surfaces.

Change of Variables Theorem

To simplify the integral, we will use the change of variables theorem. This theorem allows us to transform the integral from one coordinate system to another. In this case, we will use spherical coordinates to transform the integral.

Let's define the following transformations:

• $x = \rho \sin \phi \cos \theta$
• $y = \rho \sin \phi \sin \theta$
• $z = \rho \cos \phi$

where $\rho$ is the radial distance, $\phi$ is the polar angle, and $\theta$ is the azimuthal angle.

Spherical Coordinates

In spherical coordinates, the region $E$ is defined as:

• $a < \rho < 2a$
• $0 < \phi < \pi$
• $0 < \theta < 2\pi$

The Jacobian of the transformation is given by:

$$J = \rho^2 \sin \phi$$

Transforming the Integral

Using the change of variables theorem, we can transform the integral as follows:

$$\int_Ez^2dxdydz = \int_{a}^{2a} \int_{0}^{\pi} \int_{0}^{2\pi} (\rho \cos \phi)^2 \rho^2 \sin \phi d\theta d\phi d\rho$$

Evaluating the Integral

Now, let's evaluate the integral. We can start by integrating with respect to $\theta$:

$$\int_{0}^{2\pi} d\theta = 2\pi$$

Next, we can integrate with respect to $\phi$:

$$\int_{0}^{\pi} \cos^2 \phi \sin \phi d\phi = \frac{1}{3}$$

Finally, we can integrate with respect to $\rho$:

$$\int_{a}^{2a} \rho^4 d\rho = \frac{a^5}{5} (2^5 - 1)$$

Simplifying the Result

Combining the results, we get:

$$\int_Ez^2dxdydz = \frac{2\pi}{3} \frac{a^5}{5} (2^5 - 1)$$

Conclusion

In this article, we used the change of variables theorem to simplify the triple integral $\int_Ez^2dxdydz$. We transformed the integral from Cartesian coordinates to spherical coordinates and evaluated the integral using the Jacobian of the transformation. The final result is a simplified expression for the integral.

Additional Information

The change of variables theorem is a powerful tool for simplifying integrals. It allows us to transform the integral from one coordinate system to another, making it easier to evaluate. In this case, we used the theorem to transform the integral from Cartesian coordinates to spherical coordinates, which simplified the evaluation of the integral.

References

• [1] "Real Analysis" by Walter Rudin
• [2] "Lebesgue Integral" by Gerald B. Folland

Glossary

• Change of variables theorem: A theorem that allows us to transform the integral from one coordinate system to another.
• Spherical coordinates: A coordinate system that uses the radial distance, polar angle, and azimuthal angle to describe a point in space.
• Jacobian: The determinant of the transformation matrix, used to transform the integral from one coordinate system to another.
  Q&A: Solving the Triple Integral =====================================

Q: What is the change of variables theorem, and how is it used in solving the triple integral?

A: The change of variables theorem is a mathematical concept that allows us to transform the integral from one coordinate system to another. In the context of solving the triple integral, we used the theorem to transform the integral from Cartesian coordinates to spherical coordinates. This simplifies the evaluation of the integral and makes it easier to solve.

Q: Why did we choose to use spherical coordinates in solving the triple integral?

A: We chose to use spherical coordinates because the region of integration, $E$, is a sphere centered at the origin with radius $2a$. Spherical coordinates are well-suited for describing spherical regions, making it easier to evaluate the integral.

Q: What is the Jacobian of the transformation, and how is it used in solving the triple integral?

A: The Jacobian of the transformation is the determinant of the transformation matrix, used to transform the integral from one coordinate system to another. In the context of solving the triple integral, we used the Jacobian to transform the integral from Cartesian coordinates to spherical coordinates. The Jacobian is used to adjust the volume element of the integral, ensuring that the integral is evaluated correctly.

Q: How do we evaluate the integral in spherical coordinates?

A: To evaluate the integral in spherical coordinates, we integrate with respect to the radial distance, $\rho$, first. Then, we integrate with respect to the polar angle, $\phi$, and finally, we integrate with respect to the azimuthal angle, $\theta$.

Q: What is the final result of the triple integral, and how is it simplified?

A: The final result of the triple integral is $\frac{2\pi}{3} \frac{a^5}{5} (2^5 - 1)$. This result is simplified by combining the results of the integrations with respect to $\rho$, $\phi$, and $\theta$.

Q: What are some common applications of the change of variables theorem in solving integrals?

A: The change of variables theorem has many applications in solving integrals, including:

• Transforming integrals from Cartesian coordinates to spherical or cylindrical coordinates
• Simplifying integrals with complex regions of integration
• Evaluating integrals with singularities or discontinuities

Q: What are some common mistakes to avoid when using the change of variables theorem?

A: Some common mistakes to avoid when using the change of variables theorem include:

• Failing to adjust the volume element of the integral correctly
• Ignoring the Jacobian of the transformation
• Not checking the domain of the transformation

Q: How can I apply the change of variables theorem in my own work?

A: To apply the change of variables theorem in your own work, follow these steps:

1. Identify the region of integration and the coordinate system used.
2. Choose a suitable transformation to simplify the integral.
3. Calculate the Jacobian of the transformation.
4. Adjust the volume element of the integral correctly5. Evaluate the integral using the transformed coordinates.

By following these steps and being mindful of the common mistakes to avoid, you can effectively apply the change of variables theorem in solving integrals.