Finding this line integral

  • #1
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Homework Statement



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

Homework Equations





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 ...
 

Answers and Replies

  • #2
Dick
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Homework Helper
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Homework Statement



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

Homework Equations





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 ...

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.
 
  • #3
1,728
13
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.

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)...
 
  • #4
Dick
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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)...

Yes it is.
 
  • #5
haruspex
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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)...
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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