For the vector field G⃗ =(yexy+3cos(3x+y))i⃗ +(xexy+cos(3x+y))j⃗ , find the line integral of G⃗ along the curve C from the origin along the x-axis to the point (3,0) and then counterclockwise around the circumference of the circle x2+y2=9 to the point (3/2‾√,3/2‾√).

Question

How would I parameterize the line integral?

Answer 1

along the line from (0,0) to (3,0), dy=0, so that part is just

∫[0,3] G(x,0) dx

For the 2nd part, use the usual polar coordinate conversion with r=3, and you have

∫[0,π/4] G(3cosθ,3sinθ) dθ

I can't tell what your radicals are supposed to be. √3/2?

Answer 2

The line integral can be calculated using the Fundamental Theorem of Line Integrals b/c

i and j vectors are integrals of e^(xy) + sin(x+y)...

Integral of that from (0,0) to ( (3/sqrt(2)) , (3/sqrt(2)) )