- #1
SithsNGiggles
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Given value of a line integral, find line integral along "different" curves
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.
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].
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!
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!
I'm not sure how to interpret that... It can easily be mistaken for sarcasm.