Brain Gains (Rapid Recall, Jivin' Generation)

  • Set up a work integral that will give the work done by $\vec F(x,y) = (x+y, x^2)$ on an object that follows the path $\vec r(t) = (t^2, t^3)$ for $t\in [-1,3] $.

Solution

We have the following:

  • $M=x+y = t^2+t^3$,
  • $N=x^2 = (t^2)^2$,
  • $dx = 2tdt$,
  • $dy = 3t^2dt$.

This gives $$\int_C Mdx+Ndy =\int_{-1}^3 \underbrace{(t^2+t^3)}_{M}\underbrace{(2tdt)}_{dx}+\underbrace{(t^2)^2}_{N}\underbrace{(3t^2dt)}_{dy} .$$

Group Problems

  1. A force given by $\vec F = (2y,-x+y)$ acts on an object as it moves along the curve $\vec r(t) =(t^2,3t+1)$ for $-1\leq t\leq 2$. Compute $\ds \int_C\vec F\cdot \frac{d\vec r}{dt}dt$ (the work done by the force along the curve).
  2. Draw the parametric curve $\vec r(t) = (3\cos t, t/\pi, 4\sin t)$ for $0\leq t\leq 6\pi$.
    • If an object follows this parametrization, state the velocity, speed, and acceleration of that object.
    • Give a vector equation of the tangent line to the curve above at $t=\pi$.
    • Set up an integral that gives the length of this curve. Just set it up.
    • At time $t=2\pi$, compute the component of the acceleration that is parallel to the velocity, and the component of the acceleration that is orthogonal to the velocity.
  3. Work on 7.3 together.
  4. Let $\vec F = (a,b,c)$ and $\vec d = (x,y,z)$. Compute the component of $\vec F$ that parallel to $\vec d$ and the component of $\vec F$ that is orthogonal to $\vec d$. [Feel free to pick specific values for $a, b, c, x, y, z$, or do this problem symbolically.]