Question

Find equations for the bounding surfaces, set up a volume integral, and evaluate the integral to obtain a volume formula for each region. Assume that \(a, b, c, r, R,\) and \(h\) are positive constants. Ellipsoid Find the volume of an ellipsoid with axes of length \(2 a\) \(2 b,\) and \(2 c\)

Short answer

Answer: The volume formula for an ellipsoid with axes of length \(2a\), \(2b\), and \(2c\) is given by \(\frac{4}{3}\pi a b c\).

Step-by-step solution

Find the ellipsoid equation

An ellipsoid with axes of length \(2a\), \(2b\), and \(2c\) has the following equation: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\]

Set up the triple integral for volume

We set up a triple integral to calculate the volume, using the ellipsoid equation in the integrand: \[{V = \iiint_{V}\,dV}\] To evaluate this integral, we need to use a coordinate transformation to simplify our integral. We'll do this by working with spherical coordinates. This involves a transformation in the form of: \[\begin{cases} x = a\,p\,sin\theta\,cos\phi \\ y = b\,p\,sin\theta\,sin\phi \\ z = c\,p\,cos\theta, \end{cases}\] where \(p\in[0,1]\), \(theta\in[0,\pi]\), and \(\phi\in[0,2\pi]\).

Find the Jacobian

For our coordinate transformation, we'll need to find the Jacobian determinant. This is given by: \[J = \begin{vmatrix} \frac{\partial x}{\partial p} & \frac{\partial x}{\partial \theta}& \frac{\partial x}{\partial \phi} \\ \frac{\partial y}{\partial p} & \frac{\partial y}{\partial \theta}& \frac{\partial y}{\partial \phi} \\ \frac{\partial z}{\partial p} & \frac{\partial z}{\partial \theta}& \frac{\partial z}{\partial \phi} \end{vmatrix}.\] Evaluating the partial derivatives, our Jacobian becomes: \[J = \begin{vmatrix} a\,sin\theta\,cos\phi & a\,p\,cos\theta\,cos\phi & -a\,p\,sin\theta\,sin\phi \\ b\,sin\theta\,sin\phi & b\,p\,cos\theta\,sin\phi & b\,p\,sin\theta\,cos\phi \\ c\,cos\theta & -c\,p\,sin\theta & 0 \end{vmatrix},\] and after calculating the determinant, we get \(J = a\,b\,c\,p^2\,sin\theta\).

Perform the triple integral

Now we can perform the triple integral with the Jacobian: \[V = \int_{0}^{1}\int_{0}^{\pi}\int_{0}^{2\pi} a\,b\,c\,p^2\,sin\theta\, dp\,d\theta\,d\phi.\] Evaluating the integral, we get: \[V = \left(\int_{0}^{1} a\,b\,c\, p^2\,dp\right) \left(\int_{0}^{\pi} sin\theta\,d\theta\right) \left(\int_{0}^{2\pi} d\phi\right).\] Calculate each integral in order: 1. \(\int_{0}^{1} a\,b\,c\, p^2\,dp = \frac{a\,b\,c}{3}\) 2. \(\int_{0}^{\pi} sin\theta\,d\theta = 2\) 3. \(\int_{0}^{2\pi} d\phi = 2\pi\) Combining those results, we get the volume formula for the ellipsoid: \[V = \frac{4}{3}\pi a b c.\]

Key concepts

Triple Integral

A triple integral is a powerful mathematical tool used to calculate volumes of three-dimensional shapes in a variety of coordinate systems.
Triple integrals are essentially an extension of definite integrals into three dimensions. They allow us to integrate a function over a three-dimensional region.
This is particularly useful when dealing with complex shapes, such as ellipsoids, where slicing the shape into smaller sections helps uncover its volume. To perform a triple integral, we take the integral of a function with respect to three different variables, often representing the three spatial coordinates: x, y, and z.

• The process might involve spherical or cylindrical coordinates, depending on the symmetry of the problem.
• We must carefully define the limits of integration for each of these variables based on the region of interest.

In the case of an ellipsoid, converting to spherical coordinates simplifies the integration process and accounts for the shape's symmetry.

Ellipsoid Equation

The ellipsoid equation describes a stretched sphere, distinguished by axes of varying lengths. This equation mathematically represents an ellipsoid centered at the origin.
For an ellipsoid with semi-axes of lengths a, b, and c along the x, y, and z axes respectively, the standard form of the equation is\[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.\]This form is analogous to the equation of a sphere, but it scales the axes to capture the elliptical nature.

• If all three axes are equal (i.e., \(a = b = c\)), the equation simplifies to that of a sphere.
• Changing the values of a, b, and c stretches the sphere along the respective axes.

Understanding this equation is crucial for setting up volume integrals and solving problems involving ellipsoids.

Jacobian Determinant

The Jacobian determinant is a scalar value crucial when changing variables in multiple integrals.
It emerges from partial derivatives of a coordinate transformation, giving vital information about how areas or volumes scale under transformation.In cases like the ellipsoid, where we transform from Cartesian to spherical coordinates, calculating the Jacobian is essential. For the ellipsoid transformation, the Jacobian determinant \(J\) is calculated as:\[J = \begin{vmatrix}\frac{\partial x}{\partial p} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial \phi} \\frac{\partial y}{\partial p} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial \phi} \\frac{\partial z}{\partial p} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial \phi}\end{vmatrix}.\]In this ellipsoid problem, calculating the Jacobian offers \(J = abc p^2 \sin\theta\), which modifies our triple integral to reflect these scalings.

Spherical Coordinates

Spherical coordinates provide a convenient way to describe points in 3D space using three variables: radial distance, polar angle, and azimuthal angle.
They are especially useful when dealing with shapes possessing radial symmetry, such as spheres or ellipsoids.

• Radial distance \(p\) measures the distance from the origin to the point in space.
• Polar angle \(\theta\) represents the angle between the point and the positive z-axis.
• Azimuthal angle \(\phi\) is the angle from the positive x-axis to the projection of the point onto the xy-plane.

For transforming Cartesian coordinates \((x, y, z)\) into spherical coordinates \((p, \theta, \phi)\), the following relations are used:\[\begin{cases}x = a\,p\,\sin\theta\,\cos\phi \y = b\,p\,\sin\theta\,\sin\phi \z = c\,p\,\cos\theta\end{cases}.\]This transformation focuses on the symmetry and simplifies integration.