Question

Let C = C1 + C2 where C1 is the quarter circle x^2+y^2=4, z=0,from (0,2,0) to (2,0,0), and where C2 is the line segment from (2,0,0) to (3,3,3). Compute the work done by the force F(x,y,z) =

Answer 1

Not much can be done without knowing what
`\mathbf F(x,y,z)` is, but at the least we can set up the integral.

First parameterize the pieces of the contour:


`C_1:\mathbf r_1(t_1)=(2\sin t_1,2\cos t_1,0)`

`C_2:\mathbf r_2(t_2)=(1-t_2)(2,0,0)+t_2(3,3,3)=(2+t_2, 3t_2, 3t_2)`

where
`0\le t_1\le\frac\pi2` and
`0\le t_2\le1`. You have


`\mathrm d\mathbf r_1=(2\cos t_1,-2\sin t_1,0)\,\mathrm dt_1`

`\mathrm d\mathbf r_2=(1,3,3)\,\mathrm dt_2`

and so the work is given by the integral


`\displaystyle\int_C\mathbf F(x,y,z)\cdot\mathrm d\mathbf r`

`=\displaystyle\int_0^(\pi/2)\mathbf F(2\sin t_1,2\cos t_1,0)\cdot(2\cos t_1,-2\sin t_1,0)\,\mathrm dt_1`

`{}\displaystyle\,\,\,\,\,\,\,\,+\int_0^1\mathbf F(2+t_2,3t_2,3t_2)\cdot(1,3,3)\,\mathrm dt_2`