March 26th, 2019, 05:52 AM  #1 
Member Joined: Apr 2017 From: India Posts: 73 Thanks: 0  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. 
March 26th, 2019, 09:55 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,531 Thanks: 1390 
$F=(y^2, 2xy,xz^2)$ $S = (x, y, x^2 + y^2)$ $S_x = (1,0,2x)$ $S_y = (0,1,2y)$ $dS = S_x \times S_y = (2x,2y,1)$ $\nabla \times F = (0,z^2, 0)$ $\nabla \times F \cdot dS = 2yz^2$ 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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calculus, multivariable 
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