Question

Evaluate the following line integralC is the portion of the unit circle r(s) = < coss , sins >, for 0 less then equal to s less then equal to 3pie/2

Answer 1

`(1)/(2)`

Explanation:

We are given that

`r(s)=<cos s,sin s>`

`0\leq s\leq (3\pi)/(2)`

Line integral=
`\int_(0)^{(3\pi)/(2)}xy ds`

Where
`x=cos s,y=sin s`

Line integral=
`\int_(0)^{(3\pi)/(2)}coss sinsds`

Line integral=
`(1)/(2)\int_(0)^{(3\pi)/(2)}(2cosssin s)ds`

Line integral=
`(1)/(2)\int_(0)^{(3\pi)/(2)}sin2s ds`

By using the formula
`2sin scos s=sin2s`

Line integral=
`(1)/(4)[-cos2s]^{(3\pi)/(2)}_(0)`

By using the formula

`\int sinx dx=-cos x`

Line integral=
`-(1)/(4)*(cos 3\pi-cos 0)=-(1)/(4)(-1-1)=(1)/(2)`