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**Killer** Compute the flux of the vector field $\displaystyle F(x,y,z)=(x,y,z)$ through the portion of the parabolic cylinder $\displaystyle z=x^2$ bounded by the planes $\displaystyle z=a^2$ with $\displaystyle a>0,\,y=0,\,y=b>0$ oriented so that component $\displaystyle z$ of the normal is negative.

I need to solve this by not using first Divergence Theorem, and then by using it.

I think by using it the triple integral (Divergenge Theorem) could be $\displaystyle \displaystyle\int_{-a}^{a}{\int_{0}^{{{a}^{2}}}{\int_{0}^{b}{3\,dy}\,d z}\,dx}.$ Is it the correct set up?

How to solve it by not using Gauss?