Find the value of the line integral where and . (Hint: If F is conservative, the integration may be easier on an alternate path.)
A) 64
B) 16
C) 0
D) 256
E) 32

Question

Find the value of the line integral  where  and   . (Hint: If F is conservative, the integration may be easier on an alternate path.)
A) 64 
B) 16 
C) 0 
D) 256 
E) 32

Quizplus · Accepted Answer

The line integral is a path-independent line integral, which means it depends only on the endpoints and not on the path taken between them. Therefore, we can choose any path we like between (0,0) and (2,2). One simple path is the straight line y=x. Along this line, we have x=y and dx=dy, so the line integral becomes:

$\int_C 4y\,dx + 4x\,dy = \int_0^2 4x\,dx + 4x\,dx = \int_0^2 8x\,dx = 16.$

Therefore, the value of the line integral is 16.

Find the Value of the Line Integral Where