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Practice Problems for Module 13

Covering Sections 10.1 and 10.2 

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. Convert the given parametric equations to Cartesian equations and find the corresponding graph. 
    1. \(\displaystyle x=-t^2 \text{ , } y=t+1 \text{ , } -3 \leq t \leq 3 \) 

      \(\displaystyle  x = -(y-1)^2 \)


      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V84


    2. \(\displaystyle x = \sqrt{t} \text{ , } y = 1 - t  \)​

      \(\displaystyle x = \sqrt{1 -y} \text{ , } x \geq 0 \)


      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V85


  2. Eliminate the parameter to obtain the Cartesian equation and find the graph given by the following.
    1. \(\displaystyle x=6\cos(\theta)\text{ , } y=7\sin(\theta) \text{ , } -\pi/2 \leq t \leq \pi/2\)​

      \(\displaystyle \frac{x}{36} + \frac{y}{49} = 1\)


      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V86


    2. \(\displaystyle x = \sin(t) \text{ , } y = \csc(t) \text{ , } 0 < t < \pi/2 \)​

      \(\displaystyle y=\frac{1}{x}\)

      Video errata: There should be an open circle at the point \((1,1)\).


      To see the full video page and find related videos, click the following link.
      WIR 20B M152 V87


  3. ​Find the exact length of the curve given by \(x = e^t + e^{-t}\), \(y= 5 - 2t\), \(0 \leq t \leq 2\).

    \(\displaystyle e^2 - e^{-2}\)


    To see the full video page and find related videos, click the following link.
    WIR 20B M152 V88


  4. Find the integral that represents the surface area obtained by rotating the following curve about the \(x\)-axis.\[x= t\cos(t), \quad y=t\sin(t), \quad 0\leq t \leq \pi / 2\]

    \(\displaystyle 2\pi\int\limits_0^{\pi/2} \! t\sin(t)\sqrt{t^2+1}\,dt\)

    Video Errata: The Surface Area formula and the answer both need to be multiplied by \(2\pi\). Also the \(t\) inside the square root should be \(t^2\).


    To see the full video page and find related videos, click the following link.
    WIR 20B M152 V89


  5. Find the exact area of the surface obtained by rotating the given curve about the \(y\)-axis.
    \[x= 9t^2, \quad y=9t - 3t^3, \quad 0 \leq t \leq 2\]

    \(\displaystyle 162\pi\left(\frac{8}{3}+\frac{32}{5} \right)\)


    To see the full video page and find related videos, click the following link.
    WIR 20B M152 V90