Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the region in the second quadrant that is inside the cardioid \(\quad r=2+2 \sin \theta\) and outside the cardioid \(r=2+2 \cos \theta\), and find its area.

Step-by-Step Solution

Answer

The area in the second quadrant between these cardioids is \( \frac{1}{2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{4}} \left[ (2+2 \sin \theta)^2 - (2+2 \cos \theta)^2 \right] d\theta \).

Step 1: Understanding the Problem

We need to sketch the region that is in the second quadrant, which is to the left of the y-axis and above the x-axis, and is inside the cardioid given by\( r = 2 + 2 \sin \theta \) and outside the cardioid \( r = 2 + 2 \cos \theta \). Then, we will find the area of this region.

Step 2: Plot the Cardioids

The equation \( r = 2 + 2 \sin \theta \) describes a cardioid with the cusp on the positive y-axis. Similarly, \( r = 2 + 2 \cos \theta \) describes another cardioid with the cusp on the positive x-axis. These curves intersect at the origin.

Step 3: Find Intersection Points

To find where these cardioids intersect, set \( r = 2 + 2 \sin \theta = 2 + 2 \cos \theta \). Solving \( \sin \theta = \cos \theta \) gives \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \). However, in the second quadrant, we consider \( \theta \) from \( \frac{\pi}{2} \) to \( \pi \).

Step 4: Determine the Limits of Integration

In the second quadrant, for \( \theta \) values from \( \frac{\pi}{2} \) to \( \frac{3\pi}{4} \), \( r = 2 + 2 \sin \theta \) is inside and \( r = 2 + 2 \cos \theta \) is outside. The limits of \( \theta \) become \( \frac{\pi}{2} \) to \( \frac{3\pi}{4} \).

Step 5: Set Up Integration for Area

The area of the region can be calculated using polar coordinates with the integral \[\frac{1}{2} \int_{\theta_1}^{\theta_2} \left[ (r_{\text{outer}}^2) - (r_{\text{inner}}^2) \right] \, d\theta \]where \( r_{\text{outer}} = 2 + 2 \sin \theta \) and \( r_{\text{inner}} = 2 + 2 \cos \theta \).

Step 6: Compute the Integral

Calculate the area: \[\frac{1}{2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{4}} \left[ (2 + 2 \sin \theta)^2 - (2 + 2 \cos \theta)^2 \right] \, d\theta.\]This integral can be evaluated by expanding the squares and integrating each term separately.

Step 7: Evaluate and Simplify

After evaluating the integral, simplify the result to find the total area of the region within the specified boundaries.

Key Concepts

Cardioid

A cardioid is a heart-shaped curve, a type of limacon, commonly seen in polar coordinates. They are defined by equations like \( r = a + a \sin \theta \) or \( r = a + a \cos \theta \). This particular shape has a cusp, which is a pointed end. In the context of the exercise, we have two cardioids:

• \( r = 2 + 2 \sin \theta \), with the cusp on the positive y-axis.
• \( r = 2 + 2 \cos \theta \), with the cusp on the positive x-axis.

The challenge is visualizing these curves and understanding their spatial relationship, as the cardioids intersect and create unique enclosed regions. Understanding the position of cusps and symmetry helps in both sketching and analyzing the area to be calculated.

Calculus

Calculus is fundamental in understanding curves and areas, especially in polar coordinates. It involves two main components: differentiation, which looks at rates of change; and integration, which finds quantities like area or volume. Integrals are particularly useful when working with irregular shapes like those formed by cardioids, which do not conform to simple geometric formulas for area. Calculus allows us to work with these curves by offering techniques to measure and manipulate their properties, making it an invaluable tool for solving complex spatial problems involving curves.

Integration

Integration is essential to find the area under curves, especially in polar coordinates. In our problem, we use the formula: \[ \frac{1}{2} \int_{\theta_1}^{\theta_2} \left[ (r_{\text{outer}}^2) - (r_{\text{inner}}^2) \right] \, d\theta \]This formula helps in calculating the area between two curves. Here are the steps involved:

• Square the functions: Calculate \((r_{\text{outer}}^2)\) and \((r_{\text{inner}}^2)\) for each cardioid.
• Subtract and integrate: Find the difference between the squared values and integrate over the given limits.
• Halve the result: Since the formula includes \(\frac{1}{2}\), ensure the area reflects the accurate calculation for the polar region.

These steps provide a step-by-step approach to harnessing integration, allowing the calculation of the complex area bounded by intersecting cardioids.

Area Calculation

Area calculation in polar coordinates can initially appear daunting. However, the process becomes manageable with systematic steps:

• Identify the region: Determine where the two cardioids overlap in the second quadrant. We find this by analyzing the intersections and ensuring the right portion aligns with the polar limits.
• Set the limits properly: Select the correct integration range from \( \frac{\pi}{2} \) to \( \frac{3\pi}{4} \) based on the problem's quadrant constraints.
• Apply the area formula: Use the appropriate area formula to compute the region's size, taking care to subtract the inner cardioid from the outer one.

These actions help visualize and compute the area efficiently, emphasizing strategic problem-solving in polar coordinates.