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March 26th, 2019, 05:52 AM   #1
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Multivariable calculus

I am unable to set the limits and reach the conclusive answer and mark the correct option. I am trying to put the the solution in cylindrical coordinates but the answer I am getting is not matching with the options. Please help with the answer and more important the steps to reach such a rigorous conclusion.
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March 26th, 2019, 09:55 AM   #2
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$F=(y^2, 2xy,xz^2)$

$S = (x, y, x^2 + y^2)$

$S_x = (1,0,2x)$
$S_y = (0,1,2y)$

$dS = S_x \times S_y = (-2x,-2y,1)$

$\nabla \times F = (0,-z^2, 0)$

$\nabla \times F \cdot dS = 2yz^2$

To do the integration we convert to cylindrical coordinates.

$\nabla \times F \cdot dS =2r \sin(\theta) r^4 = 2r^5 \sin(\theta)$

$\displaystyle \int_0^{\frac \pi 2}\int_0^1 2r^5 \sin(\theta)~r ~dr~d\theta = \dfrac 2 7$
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