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Given value of a line integral, find line integral along different curves

  1. Sep 24, 2012 #1
    Given value of a line integral, find line integral along "different" curves

    1. The problem statement, all variables and given/known data
    I think I've got this figured out, so I'm just checking my answers:

    Suppose that

    [itex] \int_\gamma \vec{F}(\vec{r}) \cdot d\vec{r} = 17 [/itex],
    where [itex]\gamma[/itex] is the oriented curve [itex]\vec{r}(t) = \cos{t} \vec{i} + \sin{t} \vec{j}[/itex],
    [itex]0 \leq t \leq \pi/2 [/itex].

    Use this to evaluate the line integrals in 1-3.

    2. Relevant equations
    1. [itex] \int_{C_1} \vec{F}(\vec{r}) \cdot d\vec{r}[/itex], where [itex]C_1[/itex] is the curve
    [itex]\vec{r}(t) = \sin{t} \vec{i} + \cos{t} \vec{j}[/itex],
    [itex]0 \leq t \leq \pi/2 [/itex].

    2. [itex] \int_{C_2} \vec{F}(\vec{r}) \cdot d\vec{r}[/itex], where [itex]C_2[/itex] is the curve
    [itex]\vec{r}(t) = t \vec{i} + \sqrt{1-t^2} \vec{j}[/itex],
    [itex]0 \leq t \leq 1 [/itex].

    3. [itex] \int_{C_3} \vec{F}(\vec{r}) \cdot d\vec{r}[/itex], where [itex]C_3[/itex] is the curve
    [itex]\vec{r}(t) = \sqrt{1-t^2} \vec{i} + t \vec{j}[/itex],
    [itex]0 \leq t \leq 1 [/itex].

    3. The attempt at a solution
    I realize that each curve [itex]\vec{r}[/itex] represents a quarter-circle in the first quadrant, and [itex]\gamma[/itex] and [itex]C_3[/itex] have a counter-clockwise direction starting at [itex](1, 0)[/itex], whereas [itex]C_1[/itex] and [itex]C_2[/itex] have a clockwise direction starting at [itex](0, 1)[/itex].

    [itex]\vec{F}[/itex] is the same in each case, right? And if so, are the answers simply
    1. -17
    2. -17
    3. 17?

    Thanks!
     
  2. jcsd
  3. Sep 24, 2012 #2

    Dick

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    Re: Given value of a line integral, find line integral along "different" curves

    Sure. What could be wrong with that?
     
  4. Sep 24, 2012 #3
    Re: Given value of a line integral, find line integral along "different" curves

    :uhh: I'm not sure how to interpret that... It can easily be mistaken for sarcasm.
     
  5. Sep 24, 2012 #4

    Dick

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    Re: Given value of a line integral, find line integral along "different" curves

    It's not. I just meant that your understanding is clear enough I can't think what to add. Suppose I should have said that.
     
  6. Sep 24, 2012 #5
    Re: Given value of a line integral, find line integral along "different" curves

    Ok. Thank you!
     
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