Question

Use the Fundamental Theorem of Line Integrals to calculate ∫c F⃗ ⋅dr⃗ exactly, if F⃗ =3x2/3i⃗ +ey/5j⃗ , and C is the quarter of the unit circle in the first quadrant, traced counterclockwise from (1,0) to (0,1).
∫c F⃗ ⋅dr⃗ =?

Answer 1

It looks like the vector field is

F(x, y) = 3x ^(2/3) i + e ^(y/5) j

Find a scalar function f such that grad f = F :

∂f/∂x = 3x ^(2/3)   =>   f(x, y) = 9/5 x ^(5/3) + g(y)

=>   ∂f/∂y = e ^(y/5) = dg/dy   =>   g(y) = 5e ^(y/5) + K

=>   f(x, y) = 9/5 x ^(5/3) + 5e ^(y/5) + K

(where K is an arbitrary constant)

By the fundamental theorem, the integral of F over the given path is

∫c F • dr = f (0, 1) - f (1, 0) = 5e ^(1/5) - 34/5