After each problem, or each part, please pass the chalk.
- Compute the integral $\ds \int t\sqrt{4+9t^2}dt$.
- A force given by $\vec F = (y,-x+y)$ acts on an object as it moves along the curve $\vec r(t) =(t,t^2)$ for $0\leq t\leq 2$. Note that this means $x=t$ and $y=t^2$, so $\vec F = (t^2,-t+t^2)$. Compute $\ds \int_C\vec F\cdot \frac{d\vec r}{dt}dt$ (the work done by the force along the curve).
- Consider the curve $C$ parametrized by $\vec r(t) = (t^2, t^3)$ for $0\leq t\leq 2$.
- Give a vector equation of the tangent line to the curve at $t=1$.
- 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.
At this point, please wander the room and help your peers as needed.
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