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. Find the volume of an ellipsoid with axes of length $2a$ $2b,$ and $2c$

Short answer

Answer: The volume of an ellipsoid with axes of length $2a$, $2b$, and $2c$ is given by the formula: $V=8abc$.

Step-by-step solution

Understanding the Ellipsoid Geometry

The given ellipsoid has axes of length $2a$, $2b$, and $2c$. A general equation for an ellipsoid centered at the origin is given by: $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ The ellipsoid geometry has a semi-major axis along the x-direction of length $a$, a semi-major axis along the y-direction of length $b$, and a semi-major axis along the z-direction of length $c$.

Set up the Volume Integral

To find the volume of the ellipsoid, we need to set up a volume integral. To do this, we integrate the equation of the ellipsoid over the entire bounding volume. Since the ellipsoid is symmetric, we can consider just one octant (positive x, y, and z values) and then multiply the result by 8. The volume integral will take the following form: $$V=8{\int }_0^a{\int }_0^b{\int }_0^c\delta V$$ where $\delta V=dxdydz$

Change of Variables

We will change the variables in the integral to make it easy to evaluate. Let $u=\frac{x}{a},$ $v=\frac{y}{b},$ and $w=\frac{z}{c}$. The differential terms will be $dx=adu,dy=bdv,$ and $dz=cdw$. When we substitute these values into the integral, we get: $$V=8{\int }_0^1{\int }_0^1{\int }_0^1(abc)dudvdw$$

Evaluate the Volume Integral

Now we can evaluate the volume integral: $$V=8abc{\int }_0^{1}du{\int }_0^{1}dv{\int }_0^{1}dw$$ The integral can be split into 3 separate integrals: $$V=8abc\left({\int }_0^{1}du\right)\left({\int }_0^{1}dv\right)\left({\int }_0^{1}dw\right)$$ Evaluating the integrals: $$V=8abc([u{]}_0^1)([v{]}_0^1)([w{]}_0^1)$$ $$V=8abc(1-0)(1-0)(1-0)$$

Volume Formula

The volume of the ellipsoid with axes of length $2a$, $2b$, and $2c$ is: $$V=8abc$$

Key concepts

Ellipsoid Geometry

Ellipsoids are fascinating three-dimensional shapes resembling a stretched sphere. Understanding their geometry can simplify calculations and provide insight into their properties. An ellipsoid has three axes, which are the lengths defining its size and shape.

For an ellipsoid centered at the origin, the equation is:

• $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$
• Where $a,b,$ and $c$ represent the semi-major axes in the x, y, and z directions, respectively.

These axes determine the extent of the ellipsoid in each spatial direction. Understanding these parameters helps in visualizing the shape and size variations of an ellipsoid, especially when the lengths of the axes change.

The length of the full axis is twice the semi-major axis, therefore the ellipsoid extends from $-a$ to $a$ along the x-axis, and similarly for the other axes. This information is foundational when setting up integrals to compute properties of the ellipsoid, such as volume.

Change of Variables

In calculus, particularly when dealing with complex shapes like ellipsoids, changing variables simplifies the integration process. The original coordinates $(x,y,z)$ can make the integral difficult, but switching to new variables can streamline the calculation.

For the ellipsoid, consider the following substitutions:

• $u=\frac{x}{a}$
• $v=\frac{y}{b}$
• $w=\frac{z}{c}$

This transforms our integral bounds from $[0,a]$ to $[0,1]$, simplifying the process significantly. The Jacobian of the transformation must also be considered. In this ellipsoid problem, the differential terms transform as follows:

• $dx=adu$
• $dy=bdv$
• $dz=cdw$

Thus, the volume element $dxdydz$ becomes $abcdudvdw$, amplifying the requirement of integrating over simple bounds but with a scaling factor of $abc$. This approach makes evaluating complex integrals feasible and manageable.

Volume Formula

Finding the volume of an ellipsoid requires setting up a volume integral based on its geometric properties. This involves integrating over its entire volume but can be simplified due to its symmetry.

The volume is calculated using an integral over an octant due to the ellipsoid's symmetric nature, and then multiplying by 8: $$V=8{\int }_0^a{\int }_0^b{\int }_0^{c}dxdydz$$

Upon employing the change of variables, the integration limits transform to $[0,1]$, and the element $dxdydz$ becomes $abcdudvdw$. Thus, the integral becomes:$$V=8abc{\int }_0^1{\int }_0^1{\int }_0^{1}dudvdw$$

This simplifies further, as each integral of $u,v,$ and $w$ from 0 to 1 yields 1, resulting in:$$V=8abc\cdot (1\cdot 1\cdot 1)=8abc$$Thus, the simple yet elegant volume formula for an ellipsoid with semi-major axes $a,b,$ and $c,$ is $V=\frac{4}{3}\pi abc$. However, in our derived version with limits from $0$ to $1$, the adjustment factor is encompassed in our approach, leading to a clean $V=8abc$. This highlights how setting up and evaluating integrals using symmetry and variable changes can simplify seemingly complex calculations.