Evaluating line integrals Evaluate the line

Chapter 13, Problem 28E

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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

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