- Consider the curve $r=2\sin 3\theta$.
- Draw the curve.
- Compute $dx$ and $dy$ in terms of $\theta$ and $d\theta$.
- Give a vector equation of the tangent line to the curve above at $\theta=\pi/6$.
- Set up an integral formula to compute the arc length of this curve. Don't compute the integral.
- Consider the integral $\ds \int_{x=0}^{x=2}\left(\int_{y=-x}^{y=x^2}dy\right)dx$.
- Shade the region described by the bounds of the integral (i.e $0\leq x\leq 2$ and $-x\leq y\leq x^2$).
- Compute the double integral.
- Now consider the change of coordinates $x=2u-v$, $y=u+2v$ and the curve $u^2+v^2=4$.
- Compute $dx$ and $dy$, and then write them in the matrix form $$ \begin{bmatrix} dx\\dy \end{bmatrix}= \begin{bmatrix} ?&?\\?&? \end{bmatrix} \begin{bmatrix} du\\dv \end{bmatrix}.$$
- A parametrization of the curve in the $uv$ plane is $u=2\cos t, v=2\sin t$. Compute $du$ and $dv$ in terms of $t$ and $dt$, and then $dx$ and $dy$ in terms of $t$ and $dt$.
- Give an equation of the tangent line to the curve in the $xy$ plane at $t=\pi/2$ (so $(u,v) = (0,2)$, or $(x,y) = (0-2,0+4)$).
- Set up an integral to compute the arc length of the curve in the $xy$ plane.
- If you finish early, then repeat the problem above with another change of coordinates of the form $x=au+bv$, $y=cu+dv$ for different values of $a,b,c,d$.
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