The exercise asks to verify Stokes' Theorem for the field $F (x, y, z) = (y ^ 2 + z, x + z ^ 2, x^2+y)$ on the surface generated by the rotation of the curve $ y = x^2 $ with $ 1 \leq x \leq 2 $ around the $x$ axis.
I start by checking that $ \text{rot}\ F (x, y, z) = (1 - 2z, 1-2x, 1-2y)$ and I notice that the parametrization of the revolution surface is $r(t, \theta) = (t,t^2\cos(\theta), t^2\sin(\theta))$ with $1 \leq t \leq 2$ and $0 \leq \theta \leq 2 \pi$.
Then $\dfrac{\partial r}{\partial t}(t, \theta) \times \dfrac{\partial r}{\partial \theta}(t, \theta) = (1, 2t \cos(\theta), 2t \sin(\theta)) \times (0, -t^2 \cos(\theta), t^2 \cos(\theta)) = (2t^3, -t^2\cos(\theta), -t^2 \sin(\theta)),$ and then, \begin{align} \iint_S \text{rot}\ F \cdot dS &= \int_{0}^{2\pi}\int_{1}^{2} \text{rot}\ F(r(t, \theta)) \cdot (2t^3,-t^2\cos(\theta), -t^2 \sin(\theta)) dt d\theta\\ & = \int_{0}^{2\pi}\int_{1}^{2} (1 - 2[t^2\sin(\theta)], 1-2t, 1-2[t^2\cos(\theta)]) \cdot (2t^3, -t^2\cos(\theta), -t^2\sin(\theta)) dt d\theta = 15 \pi. \end{align} On the other hand, the $ S $ surface boundary is formed by two circles $C_1$ and $C_2$, $C_1$ centered on $(1,0,0)$ with radius $1$, and $C_2$ centered on $(2,0,0)$ with radius $2$, both located in planes parallel to the YZ plane. The parametrizations of $ C_1 $ and $ C_2 $ are respectively $r_1(t) = (1, \cos(t), \sin(t))$ and $r_2(t) = (2, \cos(t), \sin(t))$ with $0 \leq \theta \leq 2 \pi$. Then \begin{align} \int_{C_1 \cup C_2}F \cdot dr &= \int_{0}^{2\pi}F(r_1(t)) \cdot r_1'(t)dt+\int_{0}^{2\pi}F(r_2(t)) \cdot r_2'(t)dt \\ &= \int_{0}^{2\pi}(\cos^2(t)+\sin(t), 1+ \sin^2(t), 1+\cos(t)) \cdot (0, -\sin(t), \cos(t)) dt+ \int_{0}^{2\pi}(4\cos^2(t)+2\sin(t), 2+ 4\sin^2(t), 4+2\cos(t)) \cdot (0, -2\sin(t), 2\cos(t)) dt = 5\pi. \end{align}
strangely, it resulted $\iint_S \text{rot}\ F \cdot dS \neq \int_{C_1 \cup C_2}F \cdot dr$. I checked the calculations and there doesn't seem to be an error in the accounts, there must be some error in some parametrization, but I'm not noticing the error, can someone point me out?