Double Integration of 4-y^2 in Bounded Region

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The discussion focuses on the double integration of the function 4 - y² over a bounded region defined by the curves y² = 2x and y² = 8 - 2x. The initial integration attempt yielded incorrect results, prompting the need for a graphical representation to clarify the limits of integration. The correct approach involves solving for y from the bounding equations to accurately determine the integration limits before proceeding with the double integral.

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Renzokuken
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1. ∬(4-y^2 )dxdy
bounded region between y^2=2x and y^2=8-2x

I took the integral 4-y^2 dx and got 4x-xy^2 from 0-4 = 16-4y^2
then I took the integral 16-4y^2 dy from -2-2 = 128/3

It is wrong and I don't know what to do.
 
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draw a picture of y^2=2x and y^2=8-2x on a graph and the limits of integration will become more clear

but to do that first solve for y
 

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