Problem 52 - Essential Calculus

Question

Volume of an Ellipsoid Consider the plane region bounded by the graph of \(\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1\) where \(a>0\) and \(b>0\). Show that the volume of the ellipsoid formed when this region revolves about the \(y\) -axis is \(\frac{4 \pi a^{2} b}{3}\).

Step-by-Step Solution

Answer

The volume of the ellipsoid formed when the given ellipse revolves about the \(y\)-axis is \(\frac{4 \pi a^{2} b}{3}\).

Step 1: Describe the ellipse

Rewrite the ellipse's equation in terms of \(x\), isolated on one side. This gives \[x = a \sqrt{1 - \left(\frac{y}{b}\right)^2}\]

Step 2: Set up the volume integral

Use the formula mentioned earlier to set up the integral. The ellipse intersects the y-axis at the points \(-b\) and \(b\), which gives the bounds of integration, and \(f(y) = a \sqrt{1 -\left(\frac{y}{b}\right)^2}\). This gives the integral \[V = \pi \int_{-b}^{b} (a \sqrt{1 -\left(\frac{y}{b}\right)^2})^2 dy\]

Step 3: Evaluate the integral

This integral simplifies to\[V = \pi a^2 \int_{-b}^{b} (1 - \left(\frac{y}{b}\right)^2) dy \]which is a basic integral. Solving for \(V\) gives \[V = \frac{4 \pi a^{2} b}{3}\]

Key Concepts

Elliptical Integral

To better understand how to calculate the volume of an ellipsoid, it's essential to delve into the concept of elliptical integrals. These integrals are special types of integrations that arise when dealing with equations involving ellipses, such as the one given in our exercise. Essentially, an elliptical integral allows us to compute areas and volumes of shapes related to ellipses.
In this specific problem, we use integrals to find the volume of a solid formed by revolving a region around an axis. The elliptical body's formula \(\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1\) vividly describes how each point on the ellipse relates to axes by scaling factors \(a\) and \(b\).
Calculating the volume involves evaluating an integral from \(-b\) to \(b\), considering how the curve upends along the axis. The process becomes less intimidating once we consider the symmetry in an ellipse – ensuring our integral is well-suited for computing areas or volumes around axes.

Revolving Solids

Revolving solids is a powerful concept that simplifies the computation of 3D shapes. It involves rotating a 2D plane figure about a given axis to create a 3D object. In our exercise, we revolve the ellipse about the \(y\)-axis. This rotation results in an ellipsoid, similar to a stretched sphere.
To perform this calculation effectively, we use a method called the disk or stonewasher method. In this method, we imagine slicing the solid into thin disks perpendicular to the axis of rotation. Each disk's radius is determined by our function \(f(y) = a \sqrt{1 -\left(\frac{y}{b}\right)^2}\).
By stacking these infinitesimally-thin disks from \(-b\) to \(b\), we essentially rebuild the volume of the ellipsoid step-by-step, which is then easy to calculate using integration. Thus, revolving solids is not just limited to ellipsoids but is versatile for any solid of revolution.

Calculus of Volumes

The calculus of volumes is a fascinating branch of mathematical calculus involving integration techniques to compute the size of 3D shapes. It becomes crucial when we want to find volumes that are irregular or non-standard, like the ellipsoid in our problem.
In our exercise, we employ the integral \(V = \pi \int_{-b}^{b} (a \sqrt{1 -\left(\frac{y}{b}\right)^2})^2 \, dy\) as the core of the computation to find the volume of the revolving solid. The integral represents a continuous sum of small volumes (disks) that make up the ellipsoid.

• The integral accounts for every thin disk's volume created by revolving the ellipse's segment around the \(y\)-axis.
• Each disk has a radius \(a \sqrt{1 - \left(\frac{y}{b}\right)^2}\) and thickness \(dy\).
• Complete evaluation gives us \(V = \frac{4 \pi a^{2} b}{3}\), confirming the ellipsoid's volume is proportional to its dimensions and the scale factor \(\pi/3\).

Using integration in the calculus of volumes simplifies complex shapes into simple computations, making it a fundamental aspect of geometry.