- 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.
- Find the volume of the box of largest volume that will fit inside the ellipsoid $\ds\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{25}=1.$
- 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 $$\int_C\vec F\cdot d\vec r = \int_a^b \vec F(\vec r(t))\cdot \frac{d\vec r}{dt}dt. $$ 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 $\int_Cf ds = \int_a^b f(\vec r(t))\left|\frac{d\vec r}{dt}\right|dt$.
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