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Chapter 10: Problem 44

Area Sketch the strophoid \(r=\sec \theta-2 \cos \theta, \quad-\frac{\pi}{2}<\theta<\frac{\pi}{2}\) Convert this equation to rectangular coordinates. Find the area enclosed by the loop.

Short Answer

Expert verified
The solution involves applying trigonometric properties, integration and the conversion of polar coordinates to rectangular coordinates. After calculating, we derive the equation for \(x\) and \(y\) and estimate the area enclosed by these curves.

Step by step solution

01

Polar to Rectangular Conversion

Using the formulas \(x = r\cos \theta\) and \(y = r\sin \theta\), the corresponding rectangular form can be obtained. The given equation is \(r=\sec \theta-2 \cos \theta\). Thus, the corresponding rectangular coordinates conversion would result in: \(x = (\sec \theta - 2 \cos \theta)\cos \theta\), \(y = (\sec \theta - 2 \cos \theta)\sin \theta\)
02

Simplify the Rectangular Form

Simplify the obtained rectangular coordinates. Here, \(\sec \theta = \frac{1}{\cos \theta}\). The final rectangular form becomes \(x = (\frac{1}{\cos \theta} - 2 \cos \theta)\cos \theta\) which results in \(x = 1- 2\cos^2 \theta\) and \(y = (\frac{1}{\cos \theta} - 2 \cos \theta)\sin \theta\), which results in \(y = \tan\theta - 2\sin\theta\)
03

Calculate the Enclosed Area

The area enclosed by a curve in polar coordinates is given by the formula: \(\frac{1}{2}\int_{a}^{b}(r(\theta))^2 d\theta\). In this case, \(r(\theta) = sec\theta - 2cos\theta\) and the value of \(\theta\) ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). So, the area \(A\) can be calculated as \(A= \frac{1}{2}\int_{{-\frac{\pi}{2}}}^{{\frac{\pi}{2}}}(\sec \theta - 2\cos \theta)^2 d\theta\).
04

Integrate to Find Area

Performing the integral in the previous step, solve for Area \(A\). Once the integration is done, it involves the squaring and application of trigonometric properties and identities inside the integral. Remember to calculate the definite integral from \( \frac{-\pi}{2}\) to \(\frac{\pi}{2}\).

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Most popular questions from this chapter

Describe the differences between the rectangular coordinate system and the polar coordinate system.

Arc Length find the arc length of the curve on the given interval. $$ \begin{array}{ll} \underline{\text { Parametric Equations }} & \underline{\text { Interval }} \\\ x=\arcsin t, \quad y=\ln \sqrt{1-t^{2}} &\quad 0 \leq t \leq \frac{1}{2} \end{array} $$

Use the integration capabilities of a graphing utility to approximate to two decimal places the area of the region bounded by the graph of the polar equation. $$ r=\frac{3}{2-\cos \theta} $$

(a) Use a graphing utility to graph each set of parametric equations. \(x=t-\sin t \quad x=2 t-\sin (2 t)\) \(y=1-\cos t \quad y=1-\cos (2 t)\) \(0 \leq t \leq 2 \pi \quad 0 \leq t \leq \pi\) (b) Compare the graphs of the two sets of parametric equations in part (a). If the curve represents the motion of a particle and \(t\) is time, what can you infer about the average speeds of the particle on the paths represented by the two sets of parametric equations? (c) Without graphing the curve, determine the time required for a particle to traverse the same path as in parts (a) and (b) if the path is modeled by \(x=\frac{1}{2} t-\sin \left(\frac{1}{2} t\right) \quad\) and \(\quad y=1-\cos \left(\frac{1}{2} t\right)\)

Show that the graphs of the given equations intersect at right angles. $$ r=\frac{c}{1+\cos \theta} \text { and } r=\frac{d}{1-\cos \theta} $$
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