- Find the work done by $\vec F = (3,4,-2)$ on an object that moves from $(0,2,1)$ to $(3,1,-5)$ along a straight line.
- The work done by the nonconstant force $\vec F = (5y,-5x)$ around the circle $\vec r(t) = (2\cos t, 2\sin t)$ is given by the integral $$
\int_C\vec F\cdot d\vec r
=\int_C(M,N)\cdot (dx,dy)
=\int_C Mdx+Ndy
= \int_a^b \vec F(\vec r(t))\cdot \frac{d\vec r}{dt}dt
.
$$
- For the given vector field and curve, state $\vec F$, $\vec r$, $d\vec r$, $M$, $N$, $x$, $y$, $dx$, and $dy$.
- Fill in the appropriate pieces of the integral above, and then compute the integral.
- For the curve $\vec r(t) = (\cos t,\sin t)$ for $0\leq t\leq \pi$, and for the function $f(x,y) = x+y$, compute the integral $\int_Cf ds = \int_a^b f(\vec r(t))\left|\frac{d\vec r}{dt}\right|dt$. Clearly state each of $x$, $y$, $dx$, $dy$, and $ds$.
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