Problem 2 - Advanced Engineering Mathematics

Question

Use a double integral in polar coordinates to find the area of the region bounded by the graphs of the given polar equations. $$ r \quad 2+\cos \theta $$

Step-by-Step Solution

Answer

The area of the region is \( 4.5\pi \).

Step 1: Convert the Polar Equation

The given polar equation is \( r = 2 + \cos(\theta) \). This is a cardioid with a loop in one direction. Polar coordinates are sufficient here and no need to convert further, let's work with this given equation.

Step 2: Identify Limits for \( \theta \)

The cardioid completes a full loop as \( \theta \) ranges from \( 0 \) to \( 2\pi \). Thus, the bounds for \( \theta \) are \( 0 \leq \theta \leq 2\pi \).

Step 3: Set Up the Double Integral

In polar coordinates, the area \( A \) can be computed by the formula \( A = \int_{\theta_1}^{\theta_2} \int_{0}^{r(\theta)} r \, dr \, d\theta \). For this problem, this becomes \( \int_{0}^{2\pi} \int_{0}^{2 + \cos \theta} r \, dr \, d\theta \).

Step 4: Integrate with respect to \( r \) First

Integrate the inner integral: \( \int_{0}^{2 + \cos \theta} r \, dr = \left[ \frac{r^2}{2} \right]_{0}^{2 + \cos \theta} = \frac{(2 + \cos \theta)^2}{2} \).

Step 5: Simplify the Expression

Expand \((2 + \cos \theta)^2\): \( (2 + \cos \theta)^2 = 4 + 4\cos \theta + \cos^2 \theta \). Hence, the expression for the inner integral becomes \( \frac{1}{2} (4 + 4\cos \theta + \cos^2 \theta) \).

Step 6: Change \( \cos^2 \theta \) Using an Identity

Use the identity \( \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \) to rewrite \( \frac{1}{2}\cos^2\theta = \frac{1}{4} + \frac{1}{4}\cos(2\theta) \).

Step 7: Simplified Inner Integral

The expression simplifies to \( 2 + 2\cos\theta + \frac{1}{4} + \frac{1}{4}\cos(2\theta) \), or \( 2.25 + 2\cos\theta + \frac{1}{4}\cos(2\theta) \).

Step 8: Integrate with respect to \( \theta \)

To compute \( \int_{0}^{2\pi} (2.25 + 2\cos\theta + \frac{1}{4}\cos(2\theta)) \, d\theta \), integrate term by term. The integral of the constant 2.25 over \( 2\pi \) is \( 4.5\pi \). The integrals \( \int_{0}^{2\pi} 2\cos\theta \, d\theta \) and \( \int_{0}^{2\pi} \frac{1}{4}\cos(2\theta) \, d\theta \) are both zero because they integrate over a full period of a cosine function.

Step 9: Calculate the Area

The area is given by the evaluation of the integrals from previous steps: \( A = 4.5\pi \).

Key Concepts

Double Integral

A double integral is a powerful tool in mathematics used to compute the volume under a surface area or to find the area in polar coordinates. In the exercise, we employed a double integral to find the area bounded by a cardioid. When working with polar coordinates, the area of a region can be calculated using the formula \[ A = \int_{\theta_1}^{\theta_2} \int_{0}^{r(\theta)} r \, dr \, d\theta \]. This formula effectively captures the radial and angular dimensions of polar coordinates. In simpler terms, you think of a double integral in polar coordinates as a way to "pile up" small pieces of area defined by radii and angles to form the entire shape that's bounded by our given equations.

Cardioid

A cardioid is a heart-shaped curve generated in polar coordinates. The name comes from its resemblance to a heart shape. It is defined by the equation \( r = 2 + \cos(\theta) \). The cardioid in our problem features a single loop, which is characteristic of the symmetry in these curves. This specific cardioid is traced out as \( \theta \) varies from \( 0 \) to \( 2\pi \) forming a complete loop. This property makes it easier to calculate the area it encloses using polar coordinates.

Area Calculation

Calculating the area in polar coordinates involves setting up and evaluating a double integral. The process begins by determining the bounds for \( \theta \) which in our cardioid example, ranges from \( 0 \) to \( 2\pi \). The main step is setting up the integral \( \int_{0}^{2\pi} \int_{0}^{2 + \cos \theta} r \, dr \, d\theta \). We first integrate with respect to \( r \), yielding the result of \( \frac{(2 + \cos \theta)^2}{2} \). The importance of expanding and simplifying this expression cannot be overlooked, as it helps ease the computation further when we integrate with respect to \( \theta \). The outer integral is evaluated by summing up these small area "slices" calculated in polar coords around the loop.

Trigonometric Identities

Trigonometric identities are crucial in simplifying expressions during integration. In this exercise, the trigonometric identity \( \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \) helped in simplifying the expression for integration. By using this identity, the computation becomes more manageable, especially when dealing with squared trigonometric functions. This simplification via identities reduces complex expressions into forms that are easier to integrate, particularly over common intervals like \( [0, 2\pi] \), where periodic behavior simplifies to zero, hence easing the overall calculation burden.