Line integral

[mashadar]

Hi, another question

line integral of -x^3dy + y^3dx over C1uC2

where C1 is circle of radius 1 counter clockwise and C2 is circle radius 2 clockwise

i set p=y^3 q=-x^3 etc

finally integrate from 0 to 2pi and r from 1 to 2 -3r^3drdt = -45pi/2, which is correct

The question is. What would have changed if instead C2 was going clockwise and C1 was going counter clockwise ? Or both clockwise/counter ?

 

[galactus]

What would have changed if instead C2 was going clockwise and C1 was going counter clockwise ?

Then, there would be opposite sign compared to the other directions.

Or both clockwise/counter ?

If they both go the same direction, then Green's Theorem would is not applicable. The region has to be to the left of the traverse direction.

If they both went the same direction, then the region would be to the right of one of the curves.