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Evaluating line integrals Evaluate the line
Chapter 13, Problem 27E(choose chapter or problem)
Evaluate the line integral \(\int_{C} \nabla \varphi \cdot d \mathbf{r}\) for the following functions \(\varphi\) and oriented curves C in two ways.
a. Use a parametric description of C and evaluate the integral directly.
b. Use the Fundamental Theorem for line integrals.
\(\varphi(x, y)=x y ; C: \mathbf{r}(t)=\langle\cos t, \sin t\rangle, \text { for } 0 \leq t \leq \pi\)
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.
a. Use a parametric description of C and evaluate the integral directly.
b. Use the Fundamental Theorem for line integrals.
\(\varphi(x, y)=x y ; C: \mathbf{r}(t)=\langle\cos t, \sin t\rangle, \text { for } 0 \leq t \leq \pi\)
ANSWER:Solution 27E