Setting Up Integrals for Regions Bounded by Two Polar Curves

Introduction

Key Concepts

Polar Coordinates Basics

Polar coordinates offer an alternative to the traditional Cartesian coordinate system by representing points in terms of their distance from a reference point (the pole) and an angle from a reference direction. A point in polar coordinates is denoted as $(r, \theta)$, where:

• r: The radial distance from the origin.
• θ: The angle measured from the positive x-axis.

Converting between polar and Cartesian coordinates is straightforward:

• From Polar to Cartesian: $x = r \cos(\theta)$, $y = r \sin(\theta)$
• From Cartesian to Polar: $r = \sqrt{x^2 + y^2}$, $θ = \tan^{-1}\left(\frac{y}{x}\right)$

Polar Curves

Polar curves are equations that define a relationship between $r$ and $θ$. Common examples include:

• Circles: $r = a$
• Roses: $r = a \cos(kθ)$ or $r = a \sin(kθ)$
• Lemniscates: $r^2 = a^2 \cos(2θ)$

These curves can often be more elegantly expressed in polar form compared to Cartesian coordinates, especially when dealing with rotational symmetry.

Finding Intersection Points

To set up integrals for regions bounded by two polar curves, it's crucial to determine their points of intersection. Equating the two polar equations allows for solving the values of $θ$ where the curves intersect: $$ r_1 = r_2 $$ Once these $θ$ values are identified, they serve as the bounds of integration for calculating the area between the curves.

Area in Polar Coordinates

The formula to find the area enclosed by a single polar curve from $θ = a$ to $θ = b$ is: $$ A = \frac{1}{2} \int_{a}^{b} r^2 \, dθ $$ When dealing with two polar curves, the area between them is found by subtracting the area under one curve from the area under the other within the same limits.

Setting Up the Integral

To set up the integral for the area between two polar curves $r_1(θ)$ and $r_2(θ)$, follow these steps:

1. Identify the Intersection Points: Find the values of $θ$ where $r_1 = r_2$ to determine the limits of integration.
2. Determine the Outer and Inner Curves: For each segment between intersection points, decide which curve is outside (has a larger $r$ value) and which is inside.
3. Set Up the Integral: The area between the curves is given by: $$ A = \frac{1}{2} \int_{a}^{b} \left( r_{\text{outer}}^2 - r_{\text{inner}}^2 \right) dθ $$

Example Problem

Consider the polar curves $r = 2 + 2\cos(θ)$ and $r = 2\cos(θ)$. To find the area of the region bounded by these two curves:

1. Find Intersection Points: $$ 2 + 2\cos(θ) = 2\cos(θ) \implies 2 = 0 $$ This implies that the curves intersect at points where $\cos(θ)$ satisfies both equations, typically leading to specific $θ$ values like $θ = 0$ and $θ = \pi$.
2. Determine Outer and Inner Curves: For $0 \leq θ \leq \pi$, $r = 2 + 2\cos(θ)$ is the outer curve, and $r = 2\cos(θ)$ is the inner curve.
3. Set Up the Integral: $$ A = \frac{1}{2} \int_{0}^{\pi} \left( (2 + 2\cos(θ))^2 - (2\cos(θ))^2 \right) dθ $$
4. Simplify and Integrate: $$ A = \frac{1}{2} \int_{0}^{\pi} \left( 4 + 8\cos(θ) + 4\cos^2(θ) - 4\cos^2(θ) \right) dθ = \frac{1}{2} \int_{0}^{\pi} (4 + 8\cos(θ)) dθ $$ $$ A = \frac{1}{2} \left[ 4θ + 8\sin(θ) \right]_0^{\pi} = \frac{1}{2} \left( 4\pi + 0 - 0 \right) = 2\pi $$

Therefore, the area of the region bounded by the two curves is $2\pi$ square units.

Handling Multiple Intersection Points

In cases where two polar curves intersect multiple times, the region of interest may consist of several distinct segments. Each segment between consecutive intersection points must be treated separately. The total area is then the sum of the areas of these individual segments: $$ A_{\text{total}} = \sum_{i=1}^{n} \frac{1}{2} \int_{θ_{i-1}}}^{θ_i} \left( r_{\text{outer},i}^2 - r_{\text{inner},i}^2 \right) dθ $$ where $θ_{i-1}$ and $θ_i$ are the bounds between intersections.

Symmetry Considerations

Exploiting symmetry can simplify the process of setting up and evaluating integrals. If the region bounded by the curves exhibits symmetry about a line (e.g., the polar axis) or the origin, you can calculate the area for one symmetric section and multiply appropriately to obtain the total area.

For example, if a region is symmetric about the polar axis, compute the area from $θ = a$ to $θ = b$ and double it: $$ A = 2 \times \frac{1}{2} \int_{a}^{b} \left( r_{\text{outer}}^2 - r_{\text{inner}}^2 \right) dθ = \int_{a}^{b} \left( r_{\text{outer}}^2 - r_{\text{inner}}^2 \right) dθ $$

Common Mistakes to Avoid

• Incorrect Limits of Integration: Always verify the points of intersection to determine accurate bounds.
• Misidentifying Outer and Inner Curves: Ensure that for each segment of integration, the outer curve has the larger $r$ value.
• Forgetting the $\frac{1}{2}$ Factor: The area formula in polar coordinates includes a $\frac{1}{2}$ multiplier.
• Neglecting Multiple Regions: When multiple regions are bounded, calculate the area for each separately and sum them.
• Ignoring Symmetry: Utilize symmetry to simplify calculations and reduce computational effort.

Advanced Applications

Setting up integrals for regions bounded by two polar curves extends to various applications in physics, engineering, and computer graphics. For instance:

• Electromagnetic Field Calculations: Determining the area within field lines.
• Engineering Design: Designing components with specific geometric constraints.
• Computer Graphics: Rendering shapes and patterns defined by polar equations.

Understanding this concept enhances problem-solving skills and provides a foundation for more complex topics in multivariable calculus and beyond.

Step-by-Step Guide to Setting Up Integrals

To systematically set up integrals for regions bounded by two polar curves, adhere to the following steps:

1. Graph the Polar Curves: Sketching the curves helps visualize the bounded region and identify intersection points.
2. Find Intersection Points: Solve $r_1(θ) = r_2(θ)$ to find the angles $θ$ where the curves intersect.
3. Determine Bounds of Integration: Use the intersection angles as the limits for $θ$ in the integral.
4. Identify Outer and Inner Curves: For each interval between intersections, determine which curve lies outside.
5. Set Up the Area Integral: Apply the area formula: $$ A = \frac{1}{2} \int_{a}^{b} \left( r_{\text{outer}}^2 - r_{\text{inner}}^2 \right) dθ $$
6. Simplify and Compute the Integral: Expand and evaluate the integral to find the area.

Another Example Problem

Find the area of the region bounded by the polar curves $r = 1 + \sin(θ)$ and $r = 2\sin(θ)$.

1. Find Intersection Points: $$ 1 + \sin(θ) = 2\sin(θ) \implies 1 = \sin(θ) \implies θ = \frac{\pi}{2} $$ The curves intersect at $θ = \frac{\pi}{2}$ and symmetrically at $θ = \frac{3\pi}{2}$.
2. Determine Outer and Inner Curves: For $0 \leq θ \leq \frac{\pi}{2}$, $r = 2\sin(θ)$ is the outer curve. For $\frac{\pi}{2} \leq θ \leq \frac{3\pi}{2}$, $r = 1 + \sin(θ)$ is the outer curve.
3. Set Up the Integrals: $$ A = \frac{1}{2} \left( \int_{0}^{\frac{\pi}{2}} (2\sin(θ))^2 dθ + \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} (1 + \sin(θ))^2 dθ \right) $$
4. Compute the Integrals: Evaluate each integral separately and sum the results.

This method ensures accurate calculation of the area between the two curves by appropriately handling the regions defined by their intersections.

Comparison Table

Summary and Key Takeaways