Rapid Recall
1. If we know $x=2u+3v$ and $y=4u+5v$, then $dA_{xy} = ? dudv$.
Solution
Note that $dx = 2du+3dv$ and $dy=4du+5dv$. We can write this as $$(dx,dy)=(2,4)du+(3,5)dv.$$ The area of the parallelogram formed by the two vectors above is $A=|2\cdot 5-3\cdot 4|=2$.
2. Draw the curve $r = 3+2\cos\theta$.
3. Set up an integral that finds the area that lies inside the curve $r = 6+3\cos\theta$ and outside a circle of radius 3.
Solution
Group problems
- Draw the region in the $xy$ plane described by $0\leq \theta \leq \pi/3$ and $0\leq r\leq 2\sin3\theta$.
- Draw the region in the $xy$ plane described by $0\leq \theta \leq \pi/4$ and $0\leq r\leq 3\sin2\theta$.
- Set up a double integral that gives the area of the region in the $xy$ plane that lies inside one petal of the rose $r=3\cos2\theta$.
- Draw and shade the region in the $xy$ plane that lies inside the curve $r=3+2\cos\theta$ and outside the curve $r=1$.
- 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 $r=1$.
- Set up a double integral that gives the area of the region in the $xy$ plane that lies inside the curve $r=2-2\cos\theta$ and inside the curve $r=2\cos\theta$.
- Set up a double integral that gives the area of the region in the $xy$ plane that lies inside the curve $r=2-2\cos\theta$ and outside the curve $r=2\cos\theta$.
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