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

Answer:

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

Step-by-step 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:

$\int _0^{\pi }\left[-costsin^3t + sintcos^3t dt\right]dt$

$= \int _0^{\pi }-sin ^3tcostdt + \int _0^{\pi }sintcos^3tdt = 0$