Problem 35 - Thomas Calculus

Question

Volume of noncircular right cylinder The region that lies in- side the cardioid \(r=1+\cos \theta\) and outside the circle \(r=1\) is the base of a solid right cylinder. The top of the cylinder lies in the plane \(z=x\) . Find the cylinder's volume.

Step-by-Step Solution

Answer

The volume of the cylinder is 0 cubic units.

Step 1: Identify the Region in Polar Coordinates

The region inside the cardioid \( r = 1 + \cos \theta \) and outside the circle \( r = 1 \) needs to be visualized. In polar coordinates, the area of interest is where \( 1 \leq r \leq 1 + \cos \theta \). The limits for \( \theta \) can be derived by equating the cardioid to the circle, i.e., setting \( 1 + \cos \theta = 1 \), leading to \( \cos \theta = 0 \). This solution gives \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \). Thus, the region's bounds are \( 0 \leq \theta \leq \pi \).

Step 2: Express Volume in Terms of Integration

The volume of the cylinder is calculated by integrating over the described region. The height of the cylinder at any point is equal to \( z = x \), where, in polar coordinates, \( x = r \cos \theta \). Therefore, the integrand for the volume will be \( r \cos \theta \). The volume \( V \) can be described by the double integral:\[ V = \int_{0}^{\pi} \int_{1}^{1+\cos \theta} r \cos \theta \, dr \, d\theta \]

Step 3: Integrate with Respect to r

Integrate the function \( r \cos \theta \) with respect to \( r \) first, from \( r = 1 \) to \( r = 1 + \cos \theta \):\[\int_{1}^{1+\cos \theta} r \cos \theta \, dr = \left[ \frac{r^2}{2} \cos \theta \right]_1^{1+\cos \theta} = \frac{(1 + \cos \theta)^2}{2} \cos \theta - \frac{1}{2} \cos \theta\]

Step 4: Simplify the Expression

Simplify the expression obtained after integrating with respect to \( r \):\[\frac{(1 + \cos \theta)^2}{2} \cos \theta - \frac{1}{2} \cos \theta = \left( 1 + 2\cos \theta + \cos^2 \theta \right) \frac{\cos \theta}{2} - \frac{1}{2} \cos \theta\]Simplifying further, we have:\[\frac{\cos \theta}{2} + \cos^2 \theta \cos \theta\]

Step 5: Integrate with Respect to θ

The next step is to integrate this expression with respect to \( \theta \) from \( 0 \) to \( \pi \):\[\int_{0}^{\pi} \left( \frac{\cos \theta}{2} + \cos^3 \theta \right) \, d\theta\]This integral is evaluated as:\[\int_{0}^{\pi} \frac{\cos \theta}{2} \, d\theta + \int_{0}^{\pi} \cos^3 \theta \, d\theta\]The first part evaluates to \( 0 \) due to symmetry, and the second part can be computed using standard trigonometric integral techniques.

Step 6: Evaluate Remaining Integrals

Evaluate the integral \( \int_{0}^{\pi} \cos^3 \theta \, d\theta \). Using the identity \( \cos^3 \theta = \cos \theta (1 - \sin^2 \theta) \), apply substitution or reduction formulas to evaluate it. After evaluation, the result will be obtained in terms of a definite value.

Step 7: Conclude the Volume

After integrating all necessary expressions, the final value results in \( 0 \). It represents the cancellation effects of the two integrals due to symmetry and periodicity in the bounds used.

Key Concepts

Polar Coordinates

Polar coordinates offer an alternative to the more common Cartesian coordinates. In polar coordinates, each point in a plane is represented by a distance from a reference point and an angle from a reference direction. The reference point is often called the origin, and the reference direction is typically the positive x-axis.

• Radius ( "): This is the distance from the origin to the point.
• Angle ( heta"): This is the angle from the reference direction to the line connecting the point to the origin.

In the given problem, we're dealing with curves described by polar equations, such as the cardioid and the circle. The boundaries of the region of integration are determined by these curves. The cardioid is given by the polar equation \( r = 1 + \cos \theta \) and provides a unique heart-shaped pattern where the circle \( r = 1 \) is a simple loop. These two curves bound the area that serves as the base of the cylinder in the problem.

Trigonometric Integrals

Trigonometric integrals are integrals involving trigonometric functions, which often come up when tackling problems in polar coordinates or when dealing with wave functions and oscillatory motion. In this exercise, we encounter trigonometric integrals while we calculate volumes.
To solve these integrals:

• We often use substitution: a simple method for transforming the integrand into a form that is easier to work with.
• Integration by parts or using identities like half-angle or power-reducing may also be necessary.

For example, evaluating integrals like \( \int \cos^3 \theta \, d\theta \) can be challenging without transforming the integrand using identities like \( \cos^3 \theta = \cos \theta (1 - \sin^2 \theta) \). Understanding these tricks often lies at the heart of solving these types of problems.

Cardioid

A cardioid is a curve shaped like a heart, coming from the Greek word for heart. In polar coordinates, it can be expressed as \( r = 1 + \cos \theta \). This form of a cardioid emerges by setting up two points on a circle and measuring the distance a point outside the circle has moved when traced over 360 degrees across a plane.

• It represents one type of limaçon, a family of curves used in various applications including acoustic engineering and antenna design.
• In the provided exercise, the cardioid forms the outer boundary of the base area of the cylinder.

Visualizing a cardioid involves plotting it over a range of \( \theta \), which in our problem is limited to domains specific to where the cardioid intersects with a circle \( r = 1 \). This heart-like boundary signifies a region between the two curves, crucial for the integration process.

Double Integral

A double integral is used to compute the volume under a surface within a bounded region in a plane. Typically represented as \( \int \int f(x,y) \, dx \, dy \), it extends the idea of a single-variable integral to functions of two variables.
In the problem, our function \( f \) is set up initially in polar coordinates as \( r \cos \theta \), representing the cylinder's height.

• The integral is expressed in terms of \( dr \, d\theta \), signifying integration over a base region defined in polar coordinates.
• This calculation was iterated twice: first with respect to \( r \) (from the circle’s radius to the cardioid’s radius), then with respect to \( \theta \) (over the range of \( 0 \) to \( \pi \)).

These steps cumulatively determine the structure's volume, and simplifying terms through trigonometrical identity helps determine the final volume result. Evaluating the double integral using symmetry and periodicity principles led to realizing that extensions like \( \int_{0}^{\pi} \cos \theta \, d\theta \) result in zero due to their symmetric cancellations, thus simplifying the evaluation process.