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Chapter 9: Problem 24

Use a graphing utility to graph the polar equations and find the area of the given region. Common interior of \(r=5-3 \sin \theta\) and \(r=5-3 \cos \theta\)

Short Answer

Expert verified
The area of the common region of the two equations is obtained by graphing the functions, determining their intersecting points and integrating across these points to find the area.

Step by step solution

01

Graphing Polar Equations

To visualize the intersection points of the two polar equations \( r = 5 - 3 \sin \theta \) and \( r = 5 - 3 \cos \theta \), a graphing utility can be used. From the graphs, determine the intersection points by finding the \(\theta\) values where the curves intersect.
02

Solve for Intersection Points

By setting the two equations equal to each other, \(5 - 3 \sin \theta = 5 - 3 \cos \theta\), it can be found that \(\sin \theta = \cos \theta\). This implies \(\theta\) = \( \frac{\pi}{4} \), \( \frac{5\pi}{4} \)
03

Computing the Area

The area of the common region is determined by integrating from one intersection point to the other. The area \( A \) in polar coordinates is given by \( A = \frac{1}{2} \int_{a}^{b} r^2 d\theta \) . Integrating for the two areas separately between the limits of \(\theta\) and then adding them gives the total area. The calculation is as follows: \( A = \frac{1}{2} \int_{0}^{\frac{\pi}{4}} (5 - 3 \sin \theta)^2 d\theta \) + \( \frac{1}{2} \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} (5 - 3 \cos \theta)^2 d\theta \)

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Most popular questions from this chapter

Find the angle \(\theta\) between the vectors. $$ \begin{array}{l} \mathbf{u}=3 \mathbf{i}+2 \mathbf{j}+\mathbf{k} \\ \mathbf{v}=2 \mathbf{i}-3 \mathbf{j} \end{array} $$

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal parallel, or neither. $$ \begin{array}{l} \mathbf{u}=\langle\cos \theta, \sin \theta,-1\rangle \\\\\mathbf{v}=\langle\sin \theta,-\cos \theta, 0\rangle \end{array} $$

Let \(A, B,\) and \(C\) be vertices of a triangle. Find \(\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A}\)

In Exercises 45 and \(46,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } \mathbf{u} \cdot \mathbf{v}=\mathbf{u} \cdot \mathbf{w} \text { and } \mathbf{u} \neq \mathbf{0}, \text { then } \mathbf{v}=\mathbf{w} $$

Find the magnitude of \(v\). Initial point of \(\mathbf{v}:(1,-3,4)\) Terminal point of \(\mathbf{v}:(1,0,-1)\)
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