Use Green's Theorem to evaluate integrate C (F · dr). (Check the orientation of the curve before applying the theorem.)

F(x, y) = vector ( sqrt(x)+4y^3, 4x^2 + sqrt(y) )

C consists of the arc of the curve y = sin(x) from (0, 0) to (pi, 0) and the line segment from (pi, 0) to (0, 0)

By Green's theorem, the line integral of F along C is equal to the integral of the curl of F (two-dimensional curl, that is) over the region bounded by C, where C is a generic path that is oriented counterclockwise. However, our C run clockwise, so we multiply the following by -1.

where R is the set

Compute the double integral:

Integrating with respect to y is trivial:

Integrating by parts with

u = x   ⇒   du = dx

dv = sin(x) dx   ⇒   v = -cos(x)

gives

while in the other integral, we have by substitution

Then our last integral evaluates to