In each problem below, set up a formula to compute the requested value. Don't actually compute the integral, but do make sure that each part of your formula has been rewritten in terms of a parameter $t$.
- The mass of a wire on the helix $\vec r(t) = (4\cos t, 3t,4\sin t)$ which has density $\delta(x,y,z)=z+3$. Remember $m=\int_C dm = \int_C \delta ds$.
- The $x$ coordinate of the centroid for the same object. Recall $\bar x = \dfrac{\int_C x ds}{\int_C ds}$.
- The $y$ coordinate of the center of mass for the same object. Recall $\bar y = \dfrac{\int_C y dm}{\int_C dm}$.
- Change the object to a wire that lies on the parabola $x=4-y^2$ for $-2\leq y\leq 1$ with density $\delta(x,y) = x^2+y^2$. Then repeat 1, 2, and 3.
- Find a function $f(x,y)$ whose derivative is $Df(x,y) = (2x+5y, 5x-4y)$, or explain why it's not possible.
- Find a function $f(x,y)$ whose derivative is $Df(x,y) = (2x+5y, 3x-4y)$, or explain why it's not possible.
|
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |