Evaluate this line integral ∫ F . dr , where F = (3x2 sin y)i + (x3 cos y)j between the origin (0,0) and the point (2,4):
(a) along straight line y = 2x
(b) along curve y = x2
The Attempt at a Solution
dr = dx i + dy j
∫ [ (3x2 sin y) i + (x3 cos y)j ] . [dx i + dy j ]
= ∫ (3x2 sin y)dx + (x3 cos y)dy
= ∫ d(x3 sin y) from [0,0] to [2,4]
Does this mean that this line integral is independent of the path taken?
(b) If the line integral is independent of path, you should get the same answer..
Does dr = dx i + dy j still hold given that it's a curve now? do i have to use the "distance along curve" formula:
dr = √[ 1 + (dy/dx)2 ] dx
I've looked up RHB textbook it says it's fine to simply use dr = dx i + dy j ...