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Problem 30 Medium Difficulty

(a) Evaluate the line integral $ \int_C \textbf{F} \cdot d\textbf{r} $, where $ \textbf{F}(x, y, z) = x
\, \textbf{i} - z \, \textbf{j} + y \, \textbf{k} $ and $ C $ is given by $ \textbf{r}(t) = 2t \, \textbf{i} + 3t \, \textbf{j} - t^2 \, \textbf{k} $, $ -1 \leqslant t \leqslant 1 $.

(b) Illustrate part (a) by using a computer to graph $ C $ and the vectors from the vector field corresponding to $ t = \pm 1 $ and $ \pm \frac{1}{2} $ (as in Figure 13).

Answer

a) $$
\int_{C} \mathbf{F} \cdot d \mathbf{r}=-2
$$
b) $$
F(r(1))=\langle 2,1,4\rangle
$$

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Video Transcript

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