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Problem 22 Hard 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) = x \textbf{i} + y \textbf{j} + xy \textbf{k} $,
$ \textbf{r}(t) = \cos t \textbf{i} + \sin t \textbf{j} + t \textbf{k} $, $ 0 \leqslant t \leqslant \pi $
Answer
$$
\int_{C} \mathbf{F} \cdot d \mathbf{r}=0
$$
Topics
Vector Calculus
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Top Calculus 3 Educators
Lily A.
Johns Hopkins University
Anna Marie V.
Campbell University
Heather Z.
Oregon State University
Caleb E.
Baylor University
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Video Transcript
So you're similar to our earlier province. We have to compute this f off our fifty and our prime tea. Then we take simply taking topper that integrated so excess co sign t. Why? Scientists x y is signed. He Coast city our priority team. Negative signed. He co sign t one So he goes from zero to pie and we we compute their thought product. So she's negative side Tico Santi plus our scientific co sign. Plus these two product this who is also cited for society. So you can someone off them. This is nothing but one over to sign Tootie on this one. The entire corruptive should be co sign t a nephew of coastline Tootie over two with goes from zero to pie. Uh, cause I to pie host co sign zero because his po sighs a periodic function off period to pirate. Do you and just say that they both host one? So this is the crucial zero and times any number is still zero so that we should get a zero at the end
Topics
Vector Calculus
Top Calculus 3 Educators
Lily A.
Johns Hopkins University
Anna Marie V.
Campbell University
Heather Z.
Oregon State University
Caleb E.
Baylor University
