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The area \(A\) of the sector of a circle of radius \(r\) and central angle \(\theta \) is \(A=\) ______________.

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True or False  The area enclosed by the graph of a polar equation and two rays that have the pole as a common vertex is found by approximating the area using sectors of a circle.

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True or False  The area \(A\) enclosed by the graph of the equation \(r=f(\theta ),\) \(r\geq 0,\) and the rays \(\theta =\alpha \) and \(\theta =\beta ,\) is given by \(A=\int_{\alpha }^{\beta }f( \theta)\, d\theta .\)

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True or False  If \(x( \theta) =r\cos \theta ,\) \(y( \theta) =r\sin \theta \) are parametric equations of the polar equation \(r=f( \theta) ,\) then \(\left( \dfrac{dx}{ d\theta }\right) ^{2}+\left( \dfrac{dy}{d\theta }\right) ^{2}=r^{2}+\left( \dfrac{dr}{d\theta }\right) ^{2}.\)

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\(r=\cos (2\theta)\)

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\(r=2 \sin (3\theta)\)

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\(r=2+2\sin \theta\)

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\(r=3-3\cos \theta\)

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\(r=3\cos \theta;\;\theta =0\;\hbox{to}\;\theta =\dfrac{\pi}{3}\)

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\(r=3\sin \theta ;\;\theta =0\;\hbox{to}\;\theta = \dfrac{\pi }{4}\)

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\(r=a\,\theta ;\;\theta =0\ \hbox{ to }\ \theta =2\pi \)

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\(r=e^{a\,\theta };\;\theta =0\ \hbox{ to }\ \theta =\dfrac{\pi }{2}\)

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\(r=1+\cos \theta \)

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\(r=2-2\sin \theta \)

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\(r=3+\sin \theta \)

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\(r=3(2-\sin \theta )\)

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\(r=8\sin ( 3\theta ) \)

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\(r=\cos (4\theta ) \)

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\(r=4\sin ( 2\theta ) \)

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\(r=5\cos ( 3\theta ) \)

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\(r^{2}=4\cos ( 2\theta ) \)

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\(r=a^{2}\cos ( 2\theta ) \)

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Inside \(r=2\sin \theta \); outside \(r=1\)

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Inside \(r=4\cos \theta \); outside \(r=2\)

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Inside \(r=\sin \theta \); outside \(r=1-\cos \theta \)

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Inside \(r^{2}=4\cos ( 2\theta ) \); outside \(r=\sqrt{2}\)

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\(r=\sin \theta ,\;0\leq \theta \leq \dfrac{\pi }{2}\)

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\(r=1+\cos \theta ,\;0\leq \theta \leq \pi \)

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\(r=e^{\theta },\;0\leq \theta \leq \pi \)

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\(r=2a\cos \theta ,\;0\leq \theta \leq \dfrac{\pi }{2}\)

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enclosed by the small loop of the limaçon \(r=1+2\cos \theta \).

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enclosed by the small loop of the limaçon \(r=1+2\sin \theta \).

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enclosed by the loop of the graph of \(r=2-\sec \theta \).

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enclosed by the loop of the graph of \(r=5+\sec \theta \).

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enclosed by \(r=2\sin ^{2}\dfrac{\theta }{2}\).

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enclosed by \(r=6\cos ^{2}\theta \).

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inside the circle \(r=8\cos \theta \) and to the right of the line \(r=2\sec \theta \).

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inside the circle \(r=10\sin \theta \) and above the line \( r=2\csc \theta \).

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outside the circle \(r=3\) and inside the cardioid \(r=2+2\cos \theta \).

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inside the circle \(r=\sin \theta \) and outside the cardioid \( r=1+\cos \theta \).

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common to the circle \(r=\cos \theta \) and the cardioid \( r=1-\cos \theta \).

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common to the circles \(r=\cos \theta \) and \(r=\sin \theta \).

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common to the inside of the cardioid \(r=1+\sin \theta \) and the outside of the cardioid \(r=1+\cos \theta \).

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common to the inside of the lemniscate \(r^{2}=8\cos ( 2\theta ) \) and the outside of the circle \(r=2.\)

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enclosed by the rays \(\theta =0\) and \(\theta =1\) and \( r=e^{-\theta },\;\ 0\leq \theta \leq 1\).

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enclosed by the rays \(\theta =0\) and \(\theta =1\) and \( r=e^{\theta },\;\ 0\leq \theta \leq 1\).

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enclosed by the rays \(\theta =1\) and \(\theta =\pi \) and \(r= \dfrac{1}{\theta },\;\ 1\leq \theta \leq \pi \).

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inside the outer loop but outside the inner loop of \(r=1+2\sin \theta \).

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Area  Find the area of the loop of the graph of \(r=\sec \theta +2\).

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Surface Area of a Sphere  Develop a formula for the surface area of a sphere of radius \(R\).

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Surface Area of a Bead  A sphere of radius \(R\) has a hole of radius \(a\lt R\) drilled through it. See the figure. The axis of the hole coincides with a diameter of the sphere.

1. Find the surface area of that part of the sphere that remains.
2. Is the area found in Problem 29, Section 9.3, the same as the area found here? If not, justify the difference.

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Surface Area of a Plug  A plug is made to repair the hole in the sphere in Problem 51.

1. What is the surface area of the plug?
2. Is the area found in Problem 30, Section 9.3, the same as the area found here? If not justify the difference.

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Area  Find the area enclosed by the loop of the strophoid \(r=\sec \theta -2\cos \theta ,\;-\dfrac{\pi}{2}\lt\theta \lt \dfrac{\pi}{2}\) as shown in the figure.

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Area

1. Graph the limaçon \(r=2-3\cos \theta .\)
2. Find the area enclosed by the inner loop. Round the answer to three decimal places.

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Show that the area enclosed by the graph of \(r\theta =a\) and the rays \(\theta =\theta _{1}\) and \(\theta =\theta _{2}\) is proportional to the difference of the radii, \(r_{1}-r_{2}\), where \(r_{1}=\dfrac{a}{\theta _{1}}\) and \(r_{2}=\dfrac{a}{\theta _{2}}.\)

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Find the area of the region that lies outside the circle \(r=1\) and inside the rose \(r=3\sin ( 3\theta )\).

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Find the area of the region that lies inside the circle \(r=2\) and outside the rose \(r=3\sin ( 2\theta ) \).