Problem 199 - Calculus Volume 3

Question

Determine a definite integral that represents the area. Region common to \(r=2\) and \(r=4 \cos \theta\)

Step-by-Step Solution

Answer

The definite integral is \( \frac{1}{2} \int_{\frac{\pi}{3}}^{\frac{5\pi}{3}} (4 - 16 \cos^2 \theta)\, d\theta \).

Step 1: Identify the Region

The region in question is bounded by the polar curves \( r = 2 \) and \( r = 4 \cos \theta \). These represent a circle and a limaçon with inner loop, respectively, in polar coordinates.

Step 2: Find Points of Intersection

Set the two equations equal to find their points of intersection: \( 2 = 4 \cos \theta \). Solving for \( \theta \), we find \( \cos \theta = \frac{1}{2} \), which gives \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \).

Step 3: Determine the Limits of Integration

The region common to both curves lies between \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \).

Step 4: Set Up the Integral for the Area

The area common to two polar curves from \( \theta = a \) to \( \theta = b \) is given by \( \frac{1}{2} \int_{a}^{b} (r_1^2 - r_2^2)\,d\theta \). Here, \( r_1 = 2 \) and \( r_2 = 4 \cos \theta \).

Step 5: Write the Integral Expression

Plugging in our limits and functions, we set up the integral as follows: \[ \frac{1}{2} \int_{\frac{\pi}{3}}^{\frac{5\pi}{3}} (4 - (4 \cos \theta)^2) \, d\theta \]

Key Concepts

Definite Integral

The definite integral is a crucial concept in calculus, especially when calculating areas under curves. Unlike indefinite integrals which represent a family of functions, definite integrals compute a single number representing the area between a curve and the x-axis over a specified interval. The mathematical symbol \(\int_{a}^{b} f(x) \, dx\) indicates:

• \(a\) and \(b\) are the bounds, marking the interval on the x-axis.
• \(f(x)\) is the function whose area under the curve you want to find.

To put it simply, the definite integral is like summing up all the tiny slices under the curve between two points.
In polar coordinates, though, integration applies to areas bounded by curves defined by functions of angles, \(\theta\). We switch to polar form when our limits and functions involve angles and radii.

Area in Polar Coordinates

Calculating areas in polar coordinates might seem tricky at first, but it follows a systematic method. In polar coordinates, instead of using rectangles, we use sectors of circles to measure regions.
When the area enclosed by a polar curve \(r = f(\theta)\) needs to be calculated, we use the formula for area \[\int_{a}^{b} \frac{1}{2} (r^2) \, d\theta\].
Here:

• \(a\) and \(b\) are the limits for \(\theta\), the angle measure starting from the positive x-axis.
• \(r\) is the radius at angle \(\theta\).

This method is straightforward but requires that the limits of \(\theta\) and intersections of curves are correctly identified. The factor of \(\frac{1}{2}\) accounts for the fact that areas are formed by sectors of a circle rather than rectangles.

Polar Curves Intersection

Finding where two polar curves intersect is key to determining the area shared between them. Intersections help define the limits of integration in polar form.
To find intersections:

• Set the radial equations equal to each other, \\(r_1 = r_2\), and solve for \(\theta\).
• Simplify to find where the curves meet, usually resulting in angles at which both radii are the same.

For example, with \(r = 2\) and \(r = 4 \cos \theta\), equating gives us \(2 = 4 \cos \theta\). Solving
this, \(\cos \theta = \frac{1}{2}\), yields \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{5\pi}{3}\). Those angles help us establish where and over what interval the area is shared.
Understanding where intersection points occur allows us to apply the right boundaries when using integrals to find areas.