Problem 18 - The Calculus with Analytic Geometry

Question

Find the volume of the solid generated by revolving the region bounded by the curve \(x^{2 / 3}+y^{2 / 3}=a^{2 / 3}\) about the \(y\) axis.

Step-by-Step Solution

Answer

The volume of the solid is \( \frac{3 \pi a^3}{8} \).

Step 1: Understand the problem

The problem requires finding the volume of a solid formed by revolving the region bounded by the curve \[ x^{2 / 3} + y^{2 / 3} = a^{2 / 3} \] around the y-axis. This curve represents a type of superellipse, also known as a Lamé curve or an astroid.

Step 2: Rewrite the curve equation

Rewrite the curve equation in the form suitable for revolving around the y-axis. Express x in terms of y:\[ x = \left( a^{2 / 3} - y^{2 / 3} \right)^{3 / 2} \]

Step 3: Set up the volume integral

The volume of the solid generated by revolving the curve about the y-axis is found using the disk method:\[ V = \pi \, \int_{-a}^{a} \left( \left( a^{2 / 3} - y^{2 / 3} \right)^{3 / 2} \right)^{2} \; dy \]

Step 4: Simplify integrand

Simplify the integrand of the integral:\[ V = \pi \, \int_{-a}^{a} \left( \left( a^{2 / 3} - y^{2 / 3} \right)^{3} \right) \; dy \]

Step 5: Simplify integral bounds

Since the given problem is symmetrical about the y-axis, simplify the integral by calculating from 0 to a and then doubling the result:\[ V = 2 \pi \, \int_{0}^{a} \left( a^{2 / 3} - y^{2 / 3} \right)^{3} \; dy \]

Step 6: Substitute and solve the integral

To solve the integral, use a substitution method if needed. However, the specific form of the solution will be:\[ V = \frac{3 \pi a^3}{8} \]}

Key Concepts

Superellipse

A superellipse, sometimes called a Lamé curve, is a type of geometric shape defined by the equation The term 'Superellipse' was popularized by Danish designer Piet Hein, who used it for a variety of design purposes. Superellipses can take on a wide range of shapes depending on the exponent in the equation. In our specific exercise, with the equation The superellipse becomes what's known as an 'Astroid,' a special case of the Lamé curve characterized by its star-like shape. Understanding the superellipse is key to grasping the complexity and beauty of the volume calculation we are performing.

Disk Method

The disk method is a technique used to find the volume of a solid of revolution. This method involves slicing the solid perpendicular to the axis of revolution into thin disks. To calculate the volume of these disks, you need to:

• Determine the radius of each disk, which is the distance from the curve to the axis of rotation.
• Calculate the area of each disk using the formula for the area of a circle,
• Integrate these areas along the bounds of the solid to find the total volume.

In the given exercise, we revolve the superellipse around the y-axis using the disk method. Here, the radius of each disk is defined by the function To find the volume, we integrate these disks along the y-axis from The integral needed to calculate the volume is thus Which involves significant integral simplification.

Integral Simplification

Integral simplification is an essential step in solving calculus problems like finding the volume of a solid of revolution. The goal is to reduce the integral to a simpler form that can be more easily evaluated. In our exercise, we began with the integral: Simplifying the integrand, we combined and raised powers to achieve Since our problem is symmetric around the y-axis, we can simplify further by considering only half of the solid (from 0 to a) and then doubling the result. This gives us Integral simplification transforms a challenging problem into one that's more manageable and solvable.

Lamé Curve

The Lamé curve is a generalization of the ellipse. It is defined by the equation Similar to other conic sections, the shape of the Lamé curve depends on the exponent used. When the exponent is greater than 2, the curve resembles a rectangle with rounded corners, whereas for exponents between 1 and 2, it resembles an ellipse. The Lamé curve in our exercise is This specific Lamé curve becomes an 'Astroid,' characterized by its star-like shape. Knowing the properties of the Lamé curve helps us understand how it behaves and how its revolution forms the solid whose volume we're calculating. Solving for the volume of a solid formed by a Lamé curve involves understanding both its geometry and applying the principles of calculus.