Question

By making two successive simple changes of variables, evaluate $$ I=\iiint x^{2} d x d y d z $$ over the ellipsoidal region $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} \leq 1 $$

Short answer

The integral evaluates to \( \frac{4 \pi a^2 bc}{15} \).

Step-by-step solution

Identify the first change of variables

The given region is an ellipsoid defined by the inequality \ \ \[\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} \leq 1\]. Consider the transformation \( u = \frac{x}{a} \), \( v = \frac{y}{b} \), and \( w = \frac{z}{c} \). This change of variables maps the ellipsoid to the unit sphere \( u^2 + v^2 + w^2 \leq 1 \).

Compute the Jacobian determinant

To transform the integral, compute the Jacobian determinant of the transformation. The Jacobian matrix is:\ \ \[ J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \ \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \ \ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{bmatrix} = \begin{bmatrix} a & 0 & 0 \ \ 0 & b & 0 \ \ 0 & 0 & c \end{bmatrix} \]. The determinant is \( abc \).

Substitute the variables and set up the new integral

By substituting \( u = \frac{x}{a} \), \( v = \frac{y}{b} \), and \( w = \frac{z}{c} \), the integral becomes: \ \ \[ I = \iiint (a^2 u^2) (abc) du dv dw \]. Since the region is now the unit sphere \( u^2 + v^2 + w^2 \leq 1 \), the new integral is: \ \ \[ I = a^2bc \iiint u^2 du dv dw \] over \( u^2 + v^2 + w^2 \leq 1 \).

Change to spherical coordinates

Next, change to spherical coordinates where \( u = \rho \sin \phi \cos \theta \), \( v = \rho \sin \phi \sin \theta \), \( w = \rho \cos \phi \). The Jacobian for spherical coordinates is \( \rho^2 \sin \phi \). Thus, the integral is: \ \ \[ I = a^2bc \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{1} (\rho \sin \phi \cos \theta)^2 \rho^2 \sin \phi d\rho d\phi d\theta \]

Evaluate the integral over the spherical coordinates

Simplify and separate the integrals: \ \ \[ I = a^2bc \int_{0}^{2\pi} \cos^2 \theta d\theta \int_{0}^{\pi} \sin^3 \phi d\phi \int_{0}^{1} \rho^4 d\rho \]. Evaluate each integral individually. \ \ \[ \int_{0}^{2\pi} \cos^2 \theta d\theta = \pi \], \ \ \[ \int_{0}^{\pi} \sin^3 \phi d\phi = \frac{4}{3} \], \ \ \[ \int_{0}^{1} \rho^4 d\rho = \frac{1}{5} \].

Combine the results

Combine the results of the integrals: \ \ \[ I = a^2bc \cdot \pi \cdot \frac{4}{3} \cdot \frac{1}{5} = \frac{4\pi a^2 bc}{15} \]

Key concepts

Jacobian Determinant

When changing variables in multiple integrals, we use the Jacobian determinant to adjust the volume element correctly. For a transformation from variables \((x, y, z)\) to \((u, v, w)\), the Jacobian matrix is constructed using partial derivatives of the old variables with respect to the new variables: \[ J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{bmatrix} \]. The determinant of this matrix gives us the scaling factor for the volume element. For instance, in the exercise, the transformation \( u = \frac{x}{a} \, v = \frac{y}{b} \, w = \frac{z}{c} \) results in a Jacobian determinant of \( abc \), signifying that each small volume element scales by this factor.

Ellipsoidal Region

Ellipsoids generalize spheres by allowing for different radii along different axes. The equation \[ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} \leq 1 \] defines an ellipsoid centered at the origin with semi-axes \(a\), \(b\), and \(c\). By transforming variables as follows: \((u,v,w) = (\frac{x}{a}, \frac{y}{b}, \frac{z}{c})\), the ellipsoidal region maps to a unit sphere where \(u^2 + v^2 + w^2 \leq 1\). This transformation simplifies the region for integration.

Spherical Coordinates

Spherical coordinates \((\rho, \phi, \theta)\) are used for problems involving symmetry about a point. These coordinates express a point in space as a distance \(\rho\) from the origin, an angle \(\phi\) from the positive z-axis, and an angle \(\theta\) from the positive x-axis in the xy-plane. The transformations are: \[ u = \rho \sin \phi \cos \theta, \ v = \rho \sin \phi \sin \theta, \ w = \rho \cos \phi \]. The Jacobian for spherical coordinates is \(\rho^2 \sin \phi\). Integrals over spherical coordinates include this factor: \[ du \, dv \, dw = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \]. This makes simplifying and evaluating integrals over spherical regions easier.

Integral Evaluation

Evaluating multiple integrals involves transforming the integral into a simpler region and using appropriate coordinate systems. For the given problem:
\[ I = \iiint x^{2} \; dx \; dy \; dz \] over an ellipsoid, we made two changes:
1. Transforming variables to map the ellipsoid to a sphere.
2. Changing to spherical coordinates to exploit symmetry.
The transformed integral: \[ I = a^2bc \iiint u^2 \; \rho^2 \sin \phi \; d\rho \; d\phi \; d\theta \] becomes: \[ I = a^2 bc \int_{0}^{2\pi} \cos^2 \theta \, d\theta \int_{0}^{\pi} \sin^3 \phi \, d\phi \int_{0}^{1} \rho^4 \, d\rho \]. Each integral is evaluated individually:

• \[ \int_{0}^{2\pi} \cos^2 \theta \, d\theta = \pi \]
• \[ \int_{0}^{\pi} \sin^3 \phi \, d\phi = \frac{4}{3} \]
• \[ \int_{0}^{1} \rho^4 \, d\rho = \frac{1}{5} \]

Combining these, the final result is: \[ I = \frac{4\pi a^2 bc}{15} \].