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Practice Problems for Module 14

Covering Sections 10.3, 10.4, and 10.5

Directions. 
The following are review problems. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. You can also follow the link for the full video page to find related videos. 
  1. Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions:
    1. \(\displaystyle 1 < r < 5, \quad 5\pi/3 \leq \theta \leq 7\pi/3 \)​


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      WIR 20B M152 V91


    2. \(\displaystyle r=\theta, \quad \theta \geq 0 \)​


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      WIR 20B M152 V92


  2. Find a polar equation for the Cartesian equation given by \(xy = 10\).​

    \(\displaystyle r^2=\dfrac{10}{\sin(\theta)\cos(\theta)}\)


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    WIR 20B M152 V93


  3. Find the Cartesian equation for the polar equation given by \(r = 5\sin(\theta)\).

    \(\displaystyle x^2 + \left(y-\frac{5}{2}\right)^2 = \frac{25}{4} \)


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    WIR 20B M152 V94


  4. Find the area of the region \(r = \sqrt{\theta}\) when \(3\pi/2 \leq \theta \leq 2\pi\).

    \(\displaystyle \dfrac{7\pi^2}{16}\)


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    WIR 20B M152 V95


  5. Find the area of the region \(r = e^{\theta/2}\) when \(\pi/6 \leq \theta \leq 3\pi/2\).

    \(\displaystyle \dfrac{e^{3\pi/2}-e^{\pi/6}}{2}\)


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    WIR 20B M152 V96


  6. Find the area enclosed by one loop of the curve \(r = \sin(12\theta)\).

    \(\displaystyle \frac{\pi}{48} \)


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    WIR 20B M152 V97


  7. Find the area of the region enclosed by the curve \(r = 3\cos(5\theta)\).

    \(\displaystyle \frac{9\pi}{4}\)


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    WIR 20B M152 V98


  8. Find the area of the region that lies within \(r = 3 - 3\sin(\theta)\) but outside of the region given by \(r = 3\).

    \(\displaystyle \frac{72-9\pi}{4}\)


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    WIR 20B M152 V99