- Consider the integral $\ds \int_{y=1}^{y=4}\left(\int_{x=-2}^{x=3}dx\right)dy$.
- Shade the region describe by the inequalities $1\leq y\leq 4$ and $-2\leq x\leq 3$.
- Compute the double integral.
- Consider the integral $\ds \int_{x=0}^{x=3}\int_{y=0}^{y=x}dydx$.
- Shade the region describe by the bounds of the integral.
- Compute the double integral.
- Compute the integral $\ds \int_{y=0}^{y=x}\int_{x=0}^{x=3}dxdy$. Why do you not get a number?
- Use a double integral to compute the area of the region between the curves $y=x^2$ and $y=x+2$.
- 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.
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