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Finding this line integral

  1. Dec 26, 2012 #1
    1. The problem statement, all variables and given/known data

    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

    2. Relevant equations



    3. The attempt at a solution

    Part (a)
    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 ...
     
  2. jcsd
  3. Dec 26, 2012 #2

    Dick

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    x and y are not independent. You can't treat y as a constant when integrating dx and vice versa. Take your first path, y=2x. r=i dx+j dy=i dx+j 2*dx. Eliminate y from the integration by putting y=2x everywhere.
     
  4. Dec 26, 2012 #3
    I know x and y are not independent, as they are bounded by y = 2x...

    But my question here is whether the integral is independent of the path taken or not since it can be reduced to ∫ d(x3 sin y)...
     
  5. Dec 26, 2012 #4

    Dick

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    Yes it is.
     
  6. Dec 26, 2012 #5

    haruspex

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    You can think of F as a field resulting from the scalar potential x3 sin y. Since that is single-valued, the field must be conservative.
     
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