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Chapter 10: Problem 50

True鈥揊alse Determine whether the statement is true or false. Explain your answer. The expression $$ \frac{1}{2} \int_{-\pi / 4}^{\pi / 4}(1-\sqrt{2} \cos \theta)^{2} d \theta $$ $$ \begin{array}{l}{\text { computes the area enclosed by the inner loop of the lima莽on }} \\ {r=1-\sqrt{2} \cos \theta .}\end{array} $$

Short Answer

Expert verified
True. The integral computes the area of the lima莽on's inner loop.

Step by step solution

01

Understanding the 尝颈尘补莽辞苍

A lima莽on is a type of polar graph shaped like a circle with either an inner loop (when the parameter under the square root is negative) or a dimple. The given lima莽on is represented by the equation \( r = 1 - \sqrt{2} \cos \theta \).
02

Determine Conditions for Inner Loop

For the lima莽on to have an inner loop, \( r \) must be negative for some \( \theta \) values because negative \( r \) values indicate crossing the origin. We check when \(1 - \sqrt{2} \cos \theta < 0\). This simplifies to \(\cos \theta > \frac{1}{\sqrt{2}}\), meaning \(\theta\) is in the intervals \([-\pi/4, \pi/4]\).
03

Area Formulation for the Inner Loop

The area enclosed by one complete loop in polar coordinates is given by \( \frac{1}{2} \int (r^2) d\theta \). We apply this to \( r = 1 - \sqrt{2} \cos \theta \), meaning the proper integral for the inner loop is \( \frac{1}{2} \int_{-\pi/4}^{\pi/4} (1-\sqrt{2} \cos \theta)^2 d\theta \).
04

Compare to Given Integral

The integral in the problem statement is \( \frac{1}{2} \int_{-\pi/4}^{\pi/4} (1-\sqrt{2} \cos \theta)^2 d\theta \), which matches the integral required to compute the area of the inner loop. Thus, the setup is correct.
05

Final Judgment

Since the integral formulation and the condition for enclosing the inner loop are accurately derived and described, the statement must be true.

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Key Concepts

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

Polar Coordinates
Polar coordinates provide a method for representing points on a plane using a radius and an angle, which are ideal for many types of curves, particularly those involving circular patterns. In the polar coordinate system, each point on the plane is identified by the distance from the origin (radius, \( r \)) and the angle from the positive x-axis (angle, \( \theta \)).
  • The angle \( \theta \) is usually measured in radians and can take any value, while the radius \( r \) is non-negative.
  • Polar coordinates are particularly advantageous when dealing with curves like circles and spirals.
Graphs such as the lima莽on, which we will discuss in more detail later, are efficiently described using polar coordinates. This method simplifies calculations such as determining areas enclosed by loops, as the integration over angles allows the shape to be easily examined from a radial perspective.
Integral Calculus
Integral calculus is fundamental in the process of finding areas under curves, a task especially crucial in polar coordinate problems. Calculating the area involves integrating over a certain range of angles, which sums up the small arc-segments from the curve to the origin within the specified angles.
  • The integral of the squared radius function, \( \int (r^2)d\theta \), calculates the area of sectors formed by the radius at small sections across \( \theta \).
  • In the case of polar coordinates, the formula for area requires a factor of \( \frac{1}{2} \), since we're calculating the area of sectors.
This makes integral calculus essential in determining the full or partial area enclosed by a polar curve, such as the areas within the polar graphs of a lima莽on.
尝颈尘补莽辞苍
尝颈尘补莽辞苍s are a family of polar curves resembling a distorted circle, and they can vary in shape from a looped form to a dimpled circle, depending on their specific equation parameters. The equation commonly takes the form \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). The presence and shape of the inner loop are determined by the parameters \( a \) and \( b \).
  • If the magnitude of \( b \) is greater than that of \( a \), the lima莽on graph will have an inner loop.
  • When \( b = a \), it produces a cardioid shape, a special kind of lima莽on.
  • For \( a > b \), the lima莽on forms a dimpled shape without an inner loop.
In this exercise, the lima莽on described by \( r = 1 - \sqrt{2} \cos \theta \) suggests an inner loop because \( -\sqrt{2} \) exceeds 1 in magnitude, thus ensuring the negative values of \( r \) for certain angles, confirming the loop.
Area Computation
Computing the area of loops in polar curves involves using specific integral formulas to add up the areas of infinitesimal sectors. For lima莽ons, this often requires focusing on the portion of the curve where the loop exists, which involves dealing with the interval where \( r \) becomes negative.
  • The basic formula for area in polar coordinates is \( \frac{1}{2} \int r^2 d\theta \), which accounts for the radial segments covering the angle \( \theta \).
  • To compute the area of the inner loop, you integrate over the angle interval where \( r < 0 \) and the loop is defined by the specific \( \theta \) values that meet this condition.
  • Therefore, the task is to find the correct \( \theta\) interval where the loop is plotted, and apply the area formula over this interval specifically.
The integral given in the exercise corresponds perfectly to the outlined area formula for the inner loop of the lima莽on discussed, specifically within \([-\pi/4, \pi/4]\), thus correctly calculating the loop area.

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