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Problem 21 Medium Difficulty

Evaluate the line integral $ \displaystyle \int_C \textbf{F} \cdot d \textbf{r} $, where $ C $ is given by the vector function $ \textbf{r} (t) $.

$ \textbf{F}(x, y, z) = \sin x \textbf{i} + \cos y \textbf{j} + xz \textbf{k} $,
$ \textbf{r}(t) = t^3 \textbf{i} - t^2 \textbf{j} + t \textbf{k} $, $ 0 \leqslant t \leqslant 1 $

Answer

$$
\int_{C} \mathbf{F} \cdot d \mathbf{r}=\frac{6}{5}-\cos (1)-\sin (1)
$$

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Video Transcript

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