Question

Find the work done by the force field f(x,y,z)=3xi+3yj+7kf(x,y,z)=3xi+3yj+7k on a particle that moves along the helix r(t)=3cos(t)i+3sin(t)j+3tk,0≤t≤2π

Answer 1

Call the path
`\mathcal C`. Then the work done by
`\mathbf f(x,y,z)` along
`\mathcal C` is given by the line integral,
`\displaystyle\int_(\mathcal C)\mathbf f(x,y,z)\cdot\mathrm d\mathbf r=\int_(t=0)^(t=2\pi)\mathbf f(x(t),y(t),z(t))\cdot(\mathrm d\mathbf r)/(\mathrm dt)\,\mathrm dt`
Swapping the
`\mathbf{ijk}` notation out for ordered component notation, I'll write

`\mathbf r(t)=(x(t),y(t),z(t))=(3\cos t,3\sin t,3t)`
so that

`(\mathrm d\mathbf r)/(\mathrm dt)=(-3\sin t,3\cos t,3)`
The line integral reduces to
`\displaystyle\int_0^1(9\cos t,9\sin t,7)\cdot(-3\sin t,3\cos t,3)\,\mathrm dt`
`=\displaystyle\int_0^1(-27\cos t\sin t+27\sin t\cos t+21)\,\mathrm dt`
`=\displaystyle21\int_0^1\mathrm dt=21`