Question

C be the circle of radius 7 centered at the origin oriented counterclockwise. Evaluate Contour integral Subscript Upper C Superscript Baseline Bold Upper F times d Bold r by parameterizing C.

Answer 1

Answer:

$\oint_cF.dr=0$

Step-by-step explanation:

Given that a circle C of radius 7

$x^{2} +y^{2} =49---(1)$

To find:

$\oint_{C}F.dr$

As NO function is given so we suppose it to be:

$F=<x,y>$

Parametric equations:

$x=rcos\theta=7cos\theta\\y=rsin\theta=7sin\theta$

Each point on circle can be then found as

$r(\theta)=<7cos\theta,7sin\theta>---(2)$

From (2) dr can be found as:

$dr=<-7sin(\theta),7cos(\theta)>d\theta---(3)$

From (2) and (3)

$\oint_cF.dr=\int_{0}^{2\pi}{<7cos\theta,7sin\theta><-7sin(\theta),7cos(\theta)>}\,d\theta\\\\\oint_cF.dr=\int_{0}^{2\pi}{<(-7cos\theta)(7sin\theta),(7sin(\theta))(7cos(\theta))>}\,d\theta\\\\\\\oint_cF.dr=\int_{0}^{2\pi}{-49cos\theta sin\theta+49sin(\theta)cos(\theta)}\,d\theta\\\\\oint_cF.dr=0$