Sections 5 and 6

This is a really good selection of diverse problems for this part of the course. Also, good choice of different difficulties. Very representative of what has been covered. My choice in the final will not be very different than this.

Good job, guys!

1. Convert to polar coordinates the following points: \( (3,\sqrt{3}), (-1,-\sqrt{3}), (-1,1)\).

2. Compute the length of the following curves (given in polar coordinates) for the interval indicated.

   \begin{align} r &= 4\sin \theta, \quad \pi/2 \leq \theta \leq 3\pi/2. \\ r &= e^{2\theta}, \quad 0 \leq \theta \leq \pi/2. \end{align}

3. Sketch the curves given as the following expressions in polar coordinates:

   \begin{align} r& =3-3 \sin⁡(\theta), & r &= \cos⁡(4\theta), & r &= 8 \cos⁡(5\theta) \end{align}

4. Sketch the regions given by the following expressions in polar coordinates:

   \begin{align} & 2 < r \leq 4, \quad 3\pi /4 < \theta \leq 5π/4 \\ & 0 \leq r \leq 3, \quad 7\pi/6 \leq \theta \leq 2\pi \end{align}

5. Convert the following curves from polar to cartesian:

   \begin{align} \frac{1}{r} &= \cot \theta \cos⁡ \theta, & r &= 10 (\cos⁡\theta+\sin⁡\theta) \end{align}

6. Find the distance between the points with polar coordinates \( (4 : 5\pi/3)\) and \( (2 : 2\pi/3) \).

7. Find the area enclosed by the indicated curves, in the given intervals:

   \begin{align} r&=7+5 \cos\theta, \quad \pi/2 \leq \theta \leq 3\pi/2. \\ r&= 6 \cos⁡(2\theta), \quad -\pi/4 \leq \theta \leq \pi/4. \end{align}