Here's the question. I was told I should be using Stokes theorem.
Here's my Work:
$\operatorname{curl} F \cdot dS$:
$$(\nabla \times F) \cdot N \, dS=(2xz^2 \hat {y}, -2 \hat{z})\cdot(18y \hat {y}, 1 \hat{z})\,dxdy$$
As for the limits, for the answer of 4 I did -1 to 1 wrt y, and -3sqrt(1-y^2) to 3sqrt(1-y^2).
For -24 I did -1 to 1, and -3 to 3, for dy and dx, respectively. Clarified below:
$$\int_{-1}^{1} \int_{-3\sqrt1-y^2}^{3\sqrt1-y^2}\,dx\,dy$$ $$\int_{-1}^{1} \int_{-3}^{3}\,dx\,dy$$
Extra:
I've been trying to get the correct answer to this problem for a while now, and I just can't seem to get it. I've looked at numerous sources for help, and most of the time I understand, but sometimes I simply do not, although at times it may be due to the source itself.
I've tried to do the question 2 different ways: One way I do it I get an answer of 4, and another way, an answer of -24, both of which I believe is incorrect because the professor has told me the answer is 6 (I remember 6, might have been 6pi, but I do remember the number 6).
The explanation and theory of Stokes theorem I understand, at least I believe I do. But there's times when I simply do not understand why the normal vector is what it is. I understand for a scalar field you do the absolute, and for a vector field you do not, but what about for Stokes? I see 2 or 3 different ways people do it, and I am confused as to how I should be doing it here. I believe that is my problem in this case, although it could be the limits itself.
Extra extra:
Again, sorry for my lack of proper formatting. I googled how to make nice formulas in stackexchange but I couldn't get a straight answer. Also, you may be asking me, why do I not convert to cylindrical coordinates? The answer is: Because I am lost. If the area over integration was a cylinder of uniform radius, I probably would have done that, but I am just confused with so many dynamics of this theorem.
Assumptions:
$z\ge0$, outwards normal vector (Confirmed with prof)