• Support PF! Buy your school textbooks, materials and every day products Here!

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

  • #1
Given value of a line integral, find line integral along "different" curves

Homework Statement


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.

Homework 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].

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!
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618


Sure. What could be wrong with that?
 
  • #3


:uhh: I'm not sure how to interpret that... It can easily be mistaken for sarcasm.
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618


:uhh: I'm not sure how to interpret that... It can easily be mistaken for sarcasm.
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.
 
  • #5


Ok. Thank you!
 

Related Threads for: Given value of a line integral, find line integral along different curves

Replies
6
Views
720
  • Last Post
Replies
2
Views
866
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
1
Views
8K
Replies
1
Views
5K
Replies
4
Views
3K
Replies
6
Views
4K
  • Last Post
Replies
12
Views
633
Top