Rapid Recall
- Which problems are you ready to present?
- Which problems did you sincerely attempt
- If we know $r=4$ and $\theta = \pi/6$, find $x$ and $y$.
- If we know $x=8$ and $y=3$, find $r$ and $\theta$.
- For the curve $\vec r(t) = (t^2+1, -t+2)$ for $0\leq t\leq 1$, the arc length is $\int_{0}^{1}\sqrt{(2t)^2+(-1)^2}dt.$ If this curve represents a rod lying in the plane, with density (mass per length) of $\delta = xy$, modify the arc length integral to give the total mass of the rod.
Group problems
After each problem, or each part, please pass the chalk.
- Consider the curve $C$ parametrized by $\vec r(t) = (t^2, t^3)$ for $0\leq t\leq 2$.
- Set up an integral that gives the length of this curve. Just set it up.
- A wire lies along the curve $C$. The density (mass per length) of the wire at a point $(x,y)$ on the curve is given by $\delta(x,y) = y+2$. Set up an integral formula that gives the total mass of the wire.
- The wire contains charged particles. The charge density (charge per length) at a point $(x,y)$ on the curve is given by the product $q(x,y)=xy$. Set up an integral formula that gives the total charge on the wire.
- Consider the ray from the origin through the point $P=(-2,2)$. What's the angle between this ray and the positive $x$ axis? What the distance from the origin to $P$?
- Plot the polar points with $(r,\theta)$ given by $(2,0)$, $(2,\pi/6)$, $(-2,\pi/6)$, $(4,\pi/2)$, $(-4,\pi/2)$.
- Give a polar equation of the curve $2x+3y=4$. (So substitute $x=r\cos\theta$ and $y=r\sin\theta$, and then solve for $r$.)
- Give a Cartesian equation of the polar curve $r=\tan\theta\sec\theta$. (Use $x=r\cos\theta$ and $y=r\sin\theta$ to work backwards.)
- We know $x=r\cos\theta$ and $y=r\sin\theta$. Compute $dx$ in terms of $r, \theta,dr, d\theta$. (If you need to, assume that everything depends on $t$, compute derivatives, then multiply by $dt$.)
- Plot the curve $r=3-2\sin\theta$.
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