Evaluating a Line Integral of a Vector Field In Exercises 45 and 46, evaluate for each
Chapter 15, Problem 45(choose chapter or problem)
In Exercises 45 and 46, evaluate \(\int_{C} F \cdot d r\) for each curve. Discuss the orientation of the curve and its effect on the value of the integral.
\(\mathbf{F}(x, y)=x^{2} \mathbf{i}+x y \mathbf{j}\)
(a) \(\mathbf{r}_{1}(t)=2 t \mathbf{i}+(t-1) \mathbf{j}, \quad 1 \leq t \leq 3\)
(b) \(\mathbf{r}_{2}(t)=2(3-t) \mathbf{i}+(2-t) \mathbf{j}, \quad 0 \leq t \leq 2\)
Text Transcription:
int_{C} F cdot dr
F(x, y) = x^{2}i + xyj
r_1 (t) = 2ti + (t - 1)j, 1 leq t leq 3
r_2 (t) = 2(3 - t)i + (2 - t)j, 0 leq t leq 2
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