Problem 34 - Calculus Early Transcendentals

Question

\(29-34\) Find the area of the region that lies inside both curves. $$ r=a \sin \theta, \quad r=b \cos \theta, \quad a>0, b>0 $$

Step-by-Step Solution

Answer

Find intersection points and integrate using polar coordinates formula.

Step 1: Understand the Problem

We have two polar equations, \(r = a \sin \theta\) and \(r = b \cos \theta\), and we need to find the area of the region common to both curves. These curves represent a circle and a cardioid, and the area of interest is where they overlap.

Step 2: Find Points of Intersection

Set \( r = a \sin \theta = b \cos \theta \) and solve for \( \theta \). This gives us \( \tan \theta = \frac{a}{b}\). The solutions are \( \theta = \arctan\left(\frac{a}{b}\right) \) and \( \theta = \pi + \arctan\left(\frac{a}{b}\right)\).

Step 3: Determine Limits of Integration

The limits of integration are between \(\theta = \arctan\left(\frac{a}{b}\right)\) and \(\theta = \pi + \arctan\left(\frac{a}{b}\right)\). These limits correspond to the angle range where the curves intersect.

Step 4: Set Up Area Integrals

To find the shared area, we use the formula for the area in polar coordinates: \(A = \frac{1}{2} \int (r_1^2 - r_2^2) \, d\theta \). Substitute \(r_1 = a \sin \theta\) and \(r_2 = b \cos \theta\) over the determined limits.

Step 5: Evaluate the Integral

Evaluate the integral between the limits: \(A = \frac{1}{2} \left[ \int_{\arctan\left(\frac{a}{b}\right)}^{\pi + \arctan\left(\frac{a}{b}\right)} (a^2 \sin^2 \theta - b^2 \cos^2 \theta) d\theta \right] \). This can be evaluated using trigonometric identities or a calculus software.

Key Concepts

Area of Regions

To find the area of a region in polar coordinates, it is important to understand how the shapes are defined. In polar coordinates, the position of a point is given by the radial distance \( r \) from the origin and the angle \( \theta \) from the positive x-axis.

The area of a region bounded by a curve \( r = f(\theta) \) is calculated using an integral in the form:
\[ A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta \]
where \( \alpha \) and \( \beta \) denote the limits of integration that define the section of the curve you are interested in. In our exercise, these limits are the angles where the two given curves intersect.

By setting up these integrals properly, you can calculate the overlap area shared by the two curves. Ensure you're subtracting the areas appropriately if the curves cross each other as the region of interest is between the intersections.

Polar Equations

Polar equations provide an interesting way to express curves compared to the Cartesian form. In our given problem, we have two polar equations: \( r = a \sin \theta \) and \( r = b \cos \theta \).

These equations define specific shapes:

• \( r = a \sin \theta \): This is a circle which is symmetric about the y-axis.
• \( r = b \cos \theta \): This represents a cardioid, which is symmetric about the x-axis when \( a = b \).

Polar equations are advantageous because they can represent these curves more naturally, and they work well for equations where symmetry around a point is important.

Each point on these curves can be described by a set of \( (r, \theta) \), making it easier to solve problems related to areas and intersections compared to Cartesian coordinates.

Trigonometric Identities

Trigonometric identities are essential tools for solving integrals involving sine and cosine functions, like in our problem. One vital identity for our solution is the Pythagorean identity:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]
This identity helps simplify expressions and calculate areas, especially when integrating expressions like \( a^2 \sin^2 \theta - b^2 \cos^2 \theta \).

Other useful identities include:

• Double angle identities: \( \sin 2\theta = 2 \sin \theta \cos \theta \) and \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).
• Power-reducing identities: \( \sin^2 \theta = \frac{1 - \cos 2\theta}{2} \) and \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \).

Using these identities, complex integrals can be simplified to expressions that are easier to evaluate, either manually or using computational tools.

Intersection Points

Finding intersection points of polar curves is crucial for computing the area of their overlap. To find where the curves \( r = a \sin \theta \) and \( r = b \cos \theta \) intersect, set the equations equal:

\[ a \sin \theta = b \cos \theta \]
This can be rearranged to calculate \( \tan \theta \):
\[ \tan \theta = \frac{a}{b} \]
The solution \( \theta = \arctan \left( \frac{a}{b} \right) \) gives you one of the intersection points. Since polar coordinates can wrap around, the second intersection can be found by adding \( \pi \):
\[ \theta = \pi + \arctan \left( \frac{a}{b} \right) \]
A proper understanding of the periodicity and behavior of trigonometric functions is key when finding these intersections in polar coordinates. These points are used as the boundaries when setting up the integral to find the area of overlap.