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Evaluating line integrals Evaluate the line
Chapter 13, Problem 28E(choose chapter or problem)
QUESTION:
Evaluate the line integral \(\int_{C} \nabla \varphi \cdot d \mathbf{r}\) for the following functions \(\varphi\) and oriented curves C in two ways.
\(\varphi(x, y)=\left(x^{2}+y^{2}\right) / 2 ; C: \mathbf{r}(t)=\langle\sin t, \cos t\rangle, \text { for } 0 \leq t \leq \pi\)
Questions & Answers
QUESTION:
Evaluate the line integral \(\int_{C} \nabla \varphi \cdot d \mathbf{r}\) for the following functions \(\varphi\) and oriented curves C in two ways.
\(\varphi(x, y)=\left(x^{2}+y^{2}\right) / 2 ; C: \mathbf{r}(t)=\langle\sin t, \cos t\rangle, \text { for } 0 \leq t \leq \pi\)
ANSWER:SolutionStep 1: Given that (x,y) =(x2 + y2)/2; C:r(t) = sin t, cos t , for 0 t