20190208

• I-Learn, Problem Set, Class Pictures

Rapid Recall

1. Which problems are you ready to present?
2. Which problems did you sincerely attempt
3. If we know $x=3u+2v$ and $y=-u+4v$, then areas in the $uv$-plane are multiplied by how much to obtain an area in the $xy$-plane.
4. Let $d\theta$ be a small angle, and let $dr$ be a small distance. Draw a region described by $\pi/3\leq \theta\leq \pi/3+d\theta$ and $4\leq r\leq 4+dr$.
5. The radian measure of an angle is defined as the quotient of two distances (which is why the measure is unitless). What are those two distances?
6. Draw the region whose area is given by the integral $\ds\int_{0}^{\pi}\int_{1}^{5+3\sin\theta}r dr d\theta$.

Group problems

1. Draw the region in the $xy$ plane whose area is given by $\ds\int_{\pi}^{2\pi}\int_{0}^{3+3\cos\theta}r dr d\theta$.
2. If we know $x=r\cos\theta$ and $y=r\sin\theta$, then small areas in the $r\theta$-plane are multiplied by how much to obtain the area of the transformed region in the $xy$-plane? (Compute differentials and then use the parallelogram area rule.)
3. Consider the polar curve $r=7$. Find the arc length of this curve for the portion of this curve with $\alpha\leq \theta\leq \beta$ (For simplicity, let $\Delta \theta = \beta - \alpha$).
4. Draw the region in the $xy$ plane described by $0\leq \theta \leq \pi$ and $0\leq r\leq 2\sin3\theta$.
5. Draw the region in the $xy$ plane described by $0\leq \theta \leq \pi$ and $0\leq r\leq 3\sin2\theta$.
6. Set up a double integral that gives the area of the region in the $xy$ plane that lies inside one petal of the rose $r=2\cos3\theta$.
7. Set up a double integral that gives the area of the region in the $xy$ plane that lies inside one petal of the rose $r=3\cos2\theta$.
8. Draw and shade the region in the $xy$ plane that lies inside the curve $r=3+2\cos\theta$ and outside the curve $r=1$.
9. Set up a double integral that gives the area of the region in the $xy$ plane that lies inside the curve $r=3+2\cos\theta$ and outside the curve $r=1$.
10. Draw the region in the $xy$ plane described by $0\leq \theta \leq \pi/4$ and $0\leq r\leq 3\sin2\theta$.

Problem Set
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• Sep24, Oct, Nov, Dec