Line integral along the curve

Question

Find the work done by a force ->F(x,y,z) = <xz, y, xy> along the curve C: ->r(t) = <4cos(t), 4sin(t), 3t> from point A(0 , 4 , 3(pi)/2) to B(-4, 0 , 3(pi) ).

Answer 1

Work is ∫ F • dr along the path This can be rewritten as

∫ dt F • (dr/dt) t ranges from pi/2 to pi

It is easy to work out what the dot product is:

There are three terms -16 t cos(t) sin(t)

16 sin(t) cos(t)

-48 sin(t) cos(t)

The three integrals can be worked out in closed form. Putting all together I got

work = 16 + 18 pi