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

[SithsNGiggles]

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

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

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

Homework Equations

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

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

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

The Attempt at a Solution

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

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

Thanks!

 

[Dick]

Sure. What could be wrong with that?

 

[SithsNGiggles]

I'm not sure how to interpret that... It can easily be mistaken for sarcasm.

 

[Dick]

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.

 

[SithsNGiggles]

Ok. Thank you!