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

`\oint_cF.dr=0\\`

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\\`