There is a part of a cylinder $y^2 + z^2 = 1$ between $x =0$ and $x = 3$ and above the xy-plane. The boundary of this surface is a curve C which is oriented counter-clockwise when viewed from above. Using Stokes' theorem, compute
$$
\oint_C \vec{F} \cdot d \vec{r}
$$
where $\vec{F}(x,y,z)= \langle y^3, 4xy^2, yz \rangle$
Stokes' theorem:
$$
\oint_C \vec{F} \cdot d \vec{r} = \int\int_S Curl \vec{F} \cdot d\vec{s}
$$
I have already calculated Curl $\vec{F}$ to be $\langle z,0,y^2 \rangle$.
I just do not know what bounds/how to calculate those bounds to integrate by and what $d\vec{s}$ is.
Best Answer
You have $F(x,y,z) = (y^3,4xy^2,yz)$. If we write $F=(P,Q,R)$, then Stokes' theorem says that $$\int_C F\cdot dr = \int_C P \ dx + Q \ dy + R \ dz = \int_S (\partial_y R - \partial_z Q) \ dydz + (\partial_z P - \partial_x R) \ dzdx + (\partial_x Q - \partial_y P) \ dxdy.$$ This simplifies to $$\int_S z \ dydz + y^2 \ dxdy. $$ From here you just need to parameterize and compute the above integral. Does that make sense? I suspect that approaching the problem using cylindrical coordinates oriented appropriately will help.
Edit: For the limits of integration, the first equation tells you that you want $0\leq r\leq 1$. For $\theta$, the key info is that we want the region to lie above the $xy$-plane. You should be able to take $0\leq\theta\leq\pi$. I'm on my phone, so this is all a bit off the cuff.