Question

Let F (x; y) = xy2i + x2y j. Evaluate ∫F.ds (from c to [infinity]) where C is the upper half of the circle of radius 1 centered at the origin orientated counterclockwise.

Answer 1

The integral
`\int F \bullets ds` is 0.

Explanation:

A parameterization of curve C can be:

X (t) = cost 0 <= t <= pi

Y (t) = sint 0 <= t <= pi

r (t) = costi + sintj

r '(t) = -sinti + costj

`Fds = [-costsin^3t + sintcos^3t] dt`

The integral
`\int F \bullets ds` is given by:

`\int _0^(\pi )\left[-costsin^3t + sintcos^3t dt\right]dt`

`= \int _0^(\pi )-sin ^3tcostdt + \int _0^(\pi )sintcos^3tdt = 0`