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Problem 29 Medium Difficulty

(a) Evaluate the line integral $ \int_C \textbf{F} \cdot d\textbf{r} $, where $ \textbf{F}(x, y) = e^{x - 1} \, \textbf{i} + xy \, \textbf{j} $ and $ C $ is given by $ \textbf{r}(t) = t^2 \, \textbf{i} + t^3 \, \textbf{j} $, $ 0 \leqslant t \leqslant 1 $.

(b) Illustrate part (a) by using a graphing calculator or computer to graph $ C $ and the vectors from the vector field corresponding to $ t = 0, 1/\sqrt{2} $, and $ 1 $ (as in Figure 13).


a) $ \int_{C} \mathbf{F} \cdot d \mathbf{r}=\frac{11}{8}-\frac{1}{e} $
b) 2.1


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Video Transcript

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