Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The area of the region between the cardioids in the second quadrant is determined by evaluating the definite integral from \(\frac{\pi}{4}\) to \(\pi\).

Step 1: Analyze cardioids' properties

The given polar equations are for cardioids. The first cardioid, \(r = 2 + 2\sin \theta\), is symmetric about the vertical axis and primarily covers the upper half-plane. The second cardioid, \(r = 2 + 2\cos \theta\), is symmetric about the horizontal axis and primarily covers the right half-plane.

Step 2: Determine the range of theta for the second quadrant

The second quadrant spans angles from \(\frac{\pi}{2}\) to \(\pi\). This range will limit our integration to only the relevant part of the cardioids occupying the second quadrant.

Step 3: Set up the integral for the area between the cardioids

The area of the region bounded by two polar curves, \(r_1(\theta)\) and \(r_2(\theta)\), is found by computing \(\frac{1}{2}\int_a^b (r_1^2 - r_2^2) \, d\theta\). Here, \(r_1 = 2 + 2\sin \theta\) and \(r_2 = 2 + 2\cos \theta\).

Step 4: Solve for the limits of integration

To find points of intersection, set the two equations equal: \(2 + 2\sin \theta = 2 + 2\cos \theta\). Simplifying, \(\sin \theta = \cos \theta\), which holds true when \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{5\pi}{4}\). For the second quadrant, we only need \(\theta = \frac{\pi}{4}\) because \(\frac{5\pi}{4}\) lies outside the range.

Step 5: Set up and solve the definite integral

Integrate \(\frac{1}{2}\int_{\frac{\pi}{4}}^{\pi} ((2 + 2\sin \theta)^2 - (2 + 2\cos \theta)^2) \, d\theta\). Expand the expressions: - \((2 + 2\sin \theta)^2 = 4 + 8\sin \theta + 4\sin^2 \theta\)- \((2 + 2\cos \theta)^2 = 4 + 8\cos \theta + 4\cos^2 \theta\)This simplifies the integral into:\[\frac{1}{2}\int_{\frac{\pi}{4}}^{\pi} (8\sin \theta - 8\cos \theta + 4\sin^2 \theta - 4\cos^2 \theta) \, d\theta\].

Step 6: Simplify and evaluate the integrals

Break the integral into parts: 1. \(\int_{} (8\sin \theta - 8\cos \theta) \, d\theta\)2. \(\int_{} 4\sin^2 \theta \, d\theta - \int_{} 4\cos^2 \theta \, d\theta\)These yield trigonometric identities and standard integration techniques for each part.After evaluating, you'll sum these parts to get the area of interest.

Key Concepts

Polar Integration

Polar integration is a technique used to calculate areas of regions when curves are defined using polar coordinates. Unlike Cartesian, polar coordinates (\(r, \theta\)) use a radius and an angle to specify the location of points.
The polar integration process involves an integral form specific to polar coordinates: \(A = \frac{1}{2}\int_a^b \left(r_1^2(\theta) - r_2^2(\theta)\right) \, d\theta \). Here, \(r_1\) and \(r_2\) are the radial distances of two curves.

• To find the area between two curves in polar form, determine \(r_1(\theta)\) and \(r_2(\theta)\).
• Identify the relevant range of \(\theta\), often found by intersection or focusing within a quadrant.
• The integral computes the net area between \(r_1\) and \(r_2\), using the given range of angles.

This method is vital for shapes often defined by radial boundaries, useful in assessing regions like sectors, rings, or complex lobes.

Cardioid Area

A cardioid is a heart-shaped curve that is often seen in polar coordinate problems. The standard forms are \(r = a + a\sin \theta\) or \(r = a + a\cos \theta\). These shapes have interesting symmetry properties:

• The \(\sin \theta\) form is vertically symmetric, covering the upper and lower halves prominently.
• The \(\cos \theta\) form is horizontally symmetric, focusing more on the left and right halves.

To determine the area between cardioids, like in our exercise, calculate where they intersect and use polar integration for the bounded area.
To find these intersection points, equate the equations: \(2 + 2\sin \theta = 2 + 2\cos \theta\), solved via trigonometric identities or direct angle substitution. The identified intersections help outline the integration limits. Once boundaries are set, you calculate the area using polar integration steps, revealing specialized regions inside the intersecting cardioids.

Trigonometric Identities

Trigonometric identities are essential tools for manipulating and simplifying expressions involving angles. In polar coordinates, these identities help transform and simplify equations, especially when integrating.
Commonly used identities include:

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(\sin \theta = \cos(\frac{\pi}{2} - \theta)\)
• \(\cos \theta = \sin(\frac{\pi}{2} - \theta)\)
• Double-angle formulas, e.g., \(\sin(2\theta) = 2\sin \theta \cos \theta\)

In our polar integration problem, you use these identities to simplify the expressions \((2 + 2\sin \theta)^2\) and \((2 + 2\cos \theta)^2\) to make integration straightforward.
By converting complex terms into simpler equivalents, trigonometric identities reduce computation complexity, paving the way for accurate integral evaluation over specific angles or ranges.