Evaluating a Line Integral for Different Parametrizations In Exercises 14, show that the

Chapter 15, Problem 1

(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.

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

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

(b) \(\mathbf{r}_{2}(\theta)=\sin \theta \mathbf{i}+\sin ^{2} \theta \mathbf{j}, \quad 0 \leq \theta \leq \frac{\pi}{2}\)

Text Transcription:

int_C F cdot dr

F(x, y) = x^{2}i + xyj

r_1 (t) = ti + t^{2}j,     0 leq t leq 1

r_2 (theta) = sin theta{i} + sin^{2} theta{j},     0 leq theta leq pi / 2