Let (mathbf{F}(x, y, z)=leftlangle z^{2}, x, yightangle), and let (C) be the curve that is given by

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Let \(\mathbf{F}(x, y, z)=\left\langle z^{2}, x, yightangle\), and let \(C\) be the curve that is given by \(\mathbf{r}(t)=\left\langle 3+5 t^{2}, 3-t^{2}, tightangle\) for \(0 \leq t \leq 2\).

(a) Calculate \(\mathbf{F}(\mathbf{r}(t))\) and \(d \mathbf{r}=\mathbf{r}^{\prime}(t) d t\).

(b) Calculate the dot product \(\mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}^{\prime}(t) d t\) and evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\).

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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