Math 214 - Multivariable Calculus - Schedule

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:

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

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).
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.
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.]