Answer :
Answer:
[tex]27\pi[/tex]
Step-by-step explanation:
The given curve is [tex]F(x,y,z)=xyi+5zk+7yk[/tex].
Let us parameterize C to obtain a vector equation for C.
We have [tex]x^2+y^2=81[/tex], [tex]x^2+y^2=9^2[/tex] .
This is the equation of a circle( a cylinder in space) with radius, r=9 units.
The parametric equation of a circle is given by:
[tex]x=r\cos t[/tex] and [tex]y=r\sin t[/tex]
This implies that:
[tex]x=9\cos t[/tex] and [tex]y=9\sin t[/tex]
From, [tex]x+z=1[/tex] , we obtain, [tex]z=1-x[/tex], [tex]\implies z=1-9\cos t[/tex]
A vector equation for C is;
[tex]r(t)=9\cos ti+9\sin tj+(1-9\cos t)k[/tex] for [tex]0\le t\le 2\pi[/tex]
[tex]\implies r'(t)=-9\sin ti+9\cos tj+9\sin tk[/tex]
Also, we have;
[tex]F(r(t))=(81\cos t \sin t)i +5(1-9\cos t)j+(63\sin t)k[/tex]
Since C is oriented counterclockwise as viewed from above, we can apply the Stokes' Theorem.
Thus, by Stokes' Theorem;
[tex]\int\limits \int\limits_Scurl F\cdot dS=\int\limits_CF\cdot dr[/tex]
[tex]\int\limits \int\limits_Scurl F\cdot dS=\int\limits_0^{2\pi}F(r(t))\cdot r'(t)dt[/tex]
[tex]=\int\limits_0^{2\pi}(-729 \cos t \sin^2t+45\cos t(1-9\cos t )+567 \sin^2t)dt=27\pi[/tex]