Answer: Evaluating a Line Integral of a Vector Field In Exercises 1124, find the value

Chapter 15, Problem 13

(choose chapter or problem)

In Exercises 11 - 24, find the value of the line integral

\(\int_{C} \mathbf{F} \cdot d \mathbf{r}\).

(Hint: If F is conservative, the integration may be easier on an alternative path.)

\(\mathbf{F}(x, y)=y \mathbf{i}-x \mathbf{j}\)

(a) \(\mathbf{r}_{1}(t)=t \mathbf{i}+t \mathbf{j}, \quad 0 \leq t \leq 1\)

(b) \(\mathbf{r}_{2}(t)=t \mathbf{i}+t^{2} \mathbf{j}, \quad 0 \leq t \leq 1\)

(c) \(\mathbf{r}_{3}(t)=t \mathbf{i}+t^{3} \mathbf{j}, \quad 0 \leq t \leq 1\)

Text Transcription:

int_C F cdot dr

F(x, y) = yi - xj

r_1 (t) = ti + tj, 0 leq t leq 1

r_2 (t) = ti + t^{2}j,     0 leq t leq 1

r_3 (t) = ti + t^{3}j,     0 leq t leq 1