?In Exercises 1 - 4, show that the value of \(\int_{C} F \cdot d r\) is the same for each parametric representation of C. \(\mathbf{F}(x, y)=y

Chapter 15, Problem 4

(choose chapter or problem)

In Exercises 1 - 4, show that the value of \(\int_{C} F \cdot d r\) is the same for each parametric representation of C.

\(\mathbf{F}(x, y)=y \mathbf{i}+x^{2} \mathbf{j}\)

(a) \(\mathbf{r}_{1}(t)=(2+t) \mathbf{i}+(3-t) \mathbf{j}, \quad 0 \leq t \leq 3\)

(b) \(\mathbf{r}_{2}(w)=(2+\ln w) \mathbf{i}+(3-\ln w) \mathbf{j}, \quad 1 \leq w \leq e^{3}\)

Text Transcription:

int_C F cdot dr

F(x, y) = yi + x^{2}j

r_1 (t) = (2 + t) i + (3 - t) j,     0 leq t leq 3

r_2 (w) = (2 + ln w) i + (3 - ln w) j,     1 leq w leq e^3