See solution: Evaluating a Line Integral of a Vector Field In Exercises 1124, find the

Chapter 15, Problem 21

(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.)

\(\mathbf{F}(x, y, z)=(2 y+x) \mathbf{i}+\left(x^{2}-z\right) \mathbf{j}+(2 y-4 z) \mathbf{k}\)

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

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

Text Transcription:

int_C F cdot dr

F(x, y, z) = (2y + x) i + (x^2 - z) j + (2y - 4z)k

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

r_2 (t) = ti + tj + (2t - 1)^{2}k,     0 leq t leq 1