Answer:
[tex]\frac{1}{2}[/tex]
Step-by-step explanation:
We are given that
[tex]r(s)=<cos s,sin s>[/tex]
[tex]0\leq s\leq \frac{3\pi}{2}[/tex]
Line integral=[tex]\int_{0}^{\frac{3\pi}{2}}xy ds[/tex]
Where [tex]x=cos s,y=sin s[/tex]
Line integral=[tex]\int_{0}^{\frac{3\pi}{2}}coss sinsds[/tex]
Line integral=[tex]\frac{1}{2}\int_{0}^{\frac{3\pi}{2}}(2cosssin s)ds[/tex]
Line integral=[tex]\frac{1}{2}\int_{0}^{\frac{3\pi}{2}}sin2s ds[/tex]
By using the formula [tex]2sin scos s=sin2s[/tex]
Line integral=[tex]\frac{1}{4}[-cos2s]^{\frac{3\pi}{2}}_{0}[/tex]
By using the formula
[tex]\int sinx dx=-cos x[/tex]
Line integral=[tex]-\frac{1}{4}\times(cos 3\pi-cos 0)=-\frac{1}{4}(-1-1)=\frac{1}{2}[/tex]
