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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) $$
F(r(1))=\langle 2,1,4\rangle
$$
Topics
Vector Calculus
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Top Calculus 3 Educators
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Campbell University
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Video Transcript
So you know the computer graphing party you and now computed integral. We first have to compute therefore far off t and that keeps us ex replaced by two tea Move Z replaced by next fifty square. So never thes t square while replaced by three t and our price off T is two three ninety If Tootie So we have to integrate Negative one toe won the dot product of these two so forty plus re t square minus sixty square. Katie So minus forty minus three. Hey, square tt Well, we don't have to compute these parts, sir, We don't have to compute this part because this's our function and our inter voice symmetry with respect to zero. So we just have the compute this which you have Ah, that t keep from that if one to one So your next one minus now minus one So is negative too
Topics
Vector Calculus
Top Calculus 3 Educators
Anna Marie V.
Campbell University
Kayleah T.
Harvey Mudd College
Kristen K.
University of Michigan - Ann Arbor
Joseph L.
Boston College
