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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).

Answer

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

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Video Transcript

So I leaves a graphing parties in calculator to you and now soaps integral. Um, so we have to compute. Therefore, off our tea we replace exp i t square Andi, why buy t Cube? So should be t score times take you on our proud till we just take a component wise which is to tea three t square. So they're thought for so integral from zero to one on DH there dot Products are too tee too. Tee square, minus one plus three seven Oh, this one by U substitution in these this season title ripped. It should just be this because when you take the room to remove the prostitute he and from Bi Chairul and the second term should be t to the eighth over. Eight times three zero to one. And you plugging one, we get it. Zero one Just one plus three over eight minus your probably zero It Ah, one over. He so we should get eleven over. Hate minus one over. Uh,