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Chapter 10: Problem 28

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the inner loop of \(r=\cos \theta-\frac{1}{2}\)

Short Answer

Expert verified
Question: Sketch the region inside the inner loop of the polar curve \(r = \cos \theta - \frac{1}{2}\), and find the area of this region. Answer: The area of the region inside the inner loop of the polar curve \(r = \cos \theta - \frac{1}{2}\) is \(\frac{\pi}{3}\).

Step by step solution

01

Graph the polar curve and identify the inner loop.

To graph the given polar curve, you can plug different values of \(\theta\) into the equation and plot the corresponding \((r, \theta)\) points on a polar coordinate graph (or use a graphing calculator). Once you have graphed the curve, you'll find that it has two loops: a larger outer loop, and a smaller inner loop. The smaller inner loop is the region we are interested in.
02

Find the range of 胃 for the inner loop.

To find the range of \(\theta\) values for the inner loop, we need to look at where the curve intersects the pole (r = 0). Equating r to 0, we have: 0 = \(\cos\theta - \frac{1}{2}\) We can solve for \(\theta\) to get the points of intersection: \(\cos\theta = \frac{1}{2}\) \(\theta = \pm \frac{\pi}{3}\) Therefore, the inner loop forms between these two points of intersection: \(\theta = -\frac{\pi}{3}\) and \(\theta = \frac{\pi}{3}\). This is the range we'll use when calculating the area.
03

Set up the integral for the area of the inner loop.

The formula for the area of a region in polar coordinates is: \(A = \frac{1}{2}\int_{\alpha}^{\beta} r^2 d\theta\) In this case, we have: \(\alpha = -\frac{\pi}{3}\), \(\beta=\frac{\pi}{3}\), and \(r = \cos\theta -\frac{1}{2}\) Plugging in these values, the integral for the area of the inner loop is: \(A = \frac{1}{2}\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} (\cos\theta - \frac{1}{2})^2 d\theta\)
04

Solve the integral and find the area.

Now, we will complete this integral: \(A = \frac{1}{2}\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} (\cos^2\theta - \cos\theta + \frac{1}{4}) d\theta\) We can break this integral into three parts: \(A = \frac{1}{2}\left[\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}\cos^2\theta d\theta - \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}\cos\theta d\theta+ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}\frac{1}{4} d\theta\right]\) Now, solve each integral separately: 1. \(\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}\cos^2\theta d\theta = \frac{1}{2}\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}(1 + \cos(2\theta))d\theta = \frac{\pi}{3}\) 2. \(\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}\cos\theta d\theta = \bigg[\sin\theta\bigg]_{-\frac{\pi}{3}}^{\frac{\pi}{3}} = 0\) 3. \(\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}\frac{1}{4} d\theta = \frac{1}{4}\bigg[\theta\bigg]_{-\frac{\pi}{3}}^{\frac{\pi}{3}} = \frac{\pi}{3}\) Thus, the area of the inner loop is: \(A = \frac{1}{2}\left(\frac{\pi}{3} + 0 + \frac{\pi}{3}\right) = \frac{1}{2}\cdot\frac{2\pi}{3} = \boxed{\frac{\pi}{3}}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integral Calculation
Integral calculation is a crucial mathematical technique used to determine areas, among other things. In polar coordinates, integrals can help find the area enclosed by a curve. The area of a region defined by a polar curve is given by \[ A = \frac{1}{2}\int_{\alpha}^{\beta} r^2 d\theta \]Here, the \(r\) represents the radial distance from the origin to the curve, while \(\theta\) signifies the angle. The bounds \(\alpha\) and \(\beta\) are the angles that define the section of the curve being examined. When dealing with polar integrals, it's essential to properly understand the distance and angles involved as they are quite different from the Cartesian system.
When evaluating an integral in a polar coordinate system, consider any symmetry in the problem to simplify computations. Additionally, breaking down complex expressions into smaller parts can aid in solving the integrals more effectively. For instance, while computing the integral of \(\left(\cos\theta - \frac{1}{2}\right)^2\), it can be expanded and addressed in parts, ultimately simplifying the calculation.
Area of a Region
Finding the area of a region in polar coordinates often centers around using the formula for polar integrals. For a curve described by its radial function \(r = f(\theta)\), the area enclosed between two angles \(\alpha\) and \(\beta\) can be calculated using the integral: \[ A = \frac{1}{2}\int_{\alpha}^{\beta} r^2 d\theta \]This formula accounts for the wedge-shaped slice of a circle, adjusting for the varying radial distance in polar form as opposed to a standard rectangle in Cartesian coordinates.
In our exercise, we determined the inner loop of the curve \(r = \cos \theta - \frac{1}{2}\). Solving \(\cos \theta - \frac{1}{2} = 0\) helped us find the angles where the curve intersects with the origin, providing the necessary boundaries for \(\theta\). By solving the integral, we obtained the area of the loop. This method offers a beautiful solution to finding areas that may otherwise be difficult to ascertain using Cartesian coordinates.
Polar Curve Sketching
Polar curve sketching is a unique practice compared to traditional graphing. In polar coordinates, each point on the graph is represented by a radius \(r\) and an angle \(\theta\). Sketching such a curve involves plotting these points at various angles, and their respective radii, to visualize the curve.
For example, the given curve \(r = \cos \theta - \frac{1}{2}\) leads to some intriguing features. By plugging values for \(\theta\) and calculating \(r\), you notice that \(r\) and \(\theta\) effectively create the desired shape. In this exercise, there were two loops, one inner and one outer, distinguished by different values of \(\theta\). Understanding where the curve intersects the polar axis (origin) is crucial for identifying these loops and their extent.
  • Use symmetry as a tool to simplify sketching.
  • Note crucial angles like \(\theta = \pm \frac{\pi}{3}\) that serve as boundaries.
Graphing calculators or software can assist in accurately visualizing complex polar equations. However, learning the manual sketching technique builds foundational skills essential for deeper understanding.

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Most popular questions from this chapter

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