Problem 40 - Thomas Calculus

Question

In polar coordinates, the average value of a function over a region \(R\) (Section 15.3\()\) is given by $$\frac{1}{\text { Area(R) }} \iint_{R} f(r, \theta) r d r d \theta$$ Volume of noncircular right cylinder The region enclosed by the lemniscate \(r^{2}=2 \cos 2 \theta\) is the base of a solid right cylinder whose top is bounded by the sphere \(z=\sqrt{2-r^{2}}\) . Find the cyl- inder's volume.

Step-by-Step Solution

Answer

The volume of the cylinder is obtained by integrating the height function \(\sqrt{2-r^2}\) over the lemniscate region using double integration in polar coordinates.

Step 1: Understand the Problem

You are asked to find the volume of a right cylinder whose base is defined by a lemniscate in polar coordinates, and the top is bounded by a sphere. This involves integrating over the region defined by the lemniscate and applying the spherical height function.

Step 2: Identify Lemniscate Region

The lemniscate is defined by the equation \(r^2 = 2 \cos 2\theta\). This defines a closed curve, and we need to determine the limits for \(r\) and \(\theta\) that describe the entire region.

Step 3: Set Up the Integration Limits for r and θ

For the lemniscate \(r^2 = 2 \cos 2\theta\), the range of \(\theta\) is from \(-\pi/4\) to \(\pi/4\). For a given \(\theta\), \(r\) ranges from \(0\) to \(\sqrt{2 \cos 2\theta}\).

Step 4: Determine the Height of Cylinder from Sphere

The height \(z\) of the cylinder at any point \((r, \theta)\) is given by the sphere equation \(z = \sqrt{2 - r^2}\). This function delineates the top surface.

Step 5: Formulate the Volume Integral

The volume \(V\) of the cylinder is given by the double integral \[ V = \int_{-\pi/4}^{\pi/4} \int_{0}^{\sqrt{2\cos 2\theta}} \sqrt{2 - r^2} \cdot r \, dr \, d\theta \] This integrates the height \(\sqrt{2 - r^2}\) over the lemniscate region, multiplying by \(r\) to account for polar coordinates.

Step 6: Evaluate the Inner Integral

Compute the inner integral \( \int_{0}^{\sqrt{2\cos 2\theta}} \sqrt{2 - r^2} \cdot r \, dr \). Use substitution methods or symmetry properties to evaluate it.

Step 7: Evaluate the Outer Integral

Once the inner integral is calculated as a function of \(\theta\), substitute back and evaluate the outer integral with respect to \(\theta\).

Step 8: Calculate and Report the Total Volume

After evaluating the necessary integrals, calculate the final numerical value that represents the volume of the cylinder.

Key Concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, often analogous to the origin in Cartesian coordinates.
In polar coordinates, the location of a point is represented as \( (r, \theta) \) where:

• \( r \): the radial coordinate, which is the distance from the pole.
• \( \theta \): the angular coordinate, which is the angle formed by the line connecting the point to the pole and the reference direction (usually the positive x-axis).

This system is particularly useful for problems involving symmetry about a point or circular patterns, like in our exercise where the shape of a lemniscate is naturally described in polar coordinates.
To convert a point \( (r, \theta) \) in polar coordinates to Cartesian coordinates (x, y), you can use these equations:

• \( x = r \cdot \cos(\theta) \)
• \( y = r \cdot \sin(\theta) \)

Understanding polar coordinates is crucial when addressing integrals over regions like lemniscates, as they simplify the equations and boundaries involved.

Double Integral

The double integral is a powerful mathematical tool used to compute the "accumulated quantity," such as area, volume, or mass, across a two-dimensional region.
It involves integrating a function first with respect to one variable and then with respect to the second variable within defined boundaries.

• For cartesian coordinates the double integral over a region \( R \) is written as \[ \iint_R f(x, y) \, dx \, dy \]
• In polar coordinates, due to the circular nature, it becomes \[ \iint_R f(r, \theta) \, r \, dr \, d\theta \] where the term \( r \) accounts for the converging nature of polar shells.

Double integrals are particularly useful here as they allow us to calculate the volume of the cylinder by integrating the height function over the lemniscate region.
After setting limits for \( r \) and \( \theta \), an outer integral and an inner integral are solved successively, building up to the total desired computation.

Lemniscate

A lemniscate is a figure-eight or infinity-shaped curve, especially notable for its symmetry about the origin in polar coordinates. It's mathematically described by equations of the form \( r^2 = a^2 \cos 2\theta \) or similar variations.
This unique shape provides an interesting region over which to integrate because of its elegance and symmetry.

• The term 'lemniscate' comes from the Latin word 'lemniscus,' meaning ribbon, due to its intertwining appearance.
• In this exercise, the lemniscate described by \( r^2 = 2 \cos 2\theta \) acts as the boundary for calculating the base of the cylinder.
• The maximum value for \( r \) within this lemniscate changes depending on \( \theta \), making it necessary to account for it when setting up integration limits.

This property makes polar coordinates a natural choice for integration, simplifying the limits and area calculations necessary to solve the problem.

Sphere Equation

The equation of a sphere in three-dimensional space is generally given by \( x^2 + y^2 + z^2 = a^2 \), where \( a \) is the radius of the sphere. When discussing surfaces or volumes involving spheres, the height related to the sphere can often be described using a rearrangement of the equation.
In our problem, the top of the cylinder is defined by a sphere's surface given by \( z = \sqrt{2 - r^2} \).

• This specific form ensures that the height at any point \( (r, \theta) \) is derived from the sphere's equation, showcasing a vertical cross-section.
• By defining \( z \) as a height function, it clearly indicates how far above the lemniscate-base each point reaches.
• This serves as an upper boundary for the volume integration process.

The boundary formed by the sphere coupled with the lemniscate base in polar coordinates results in the setup needed to solve for the entire volume of the cylinder properly.