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Find the double integral

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1. Use a double integral to find the area of the region bounded by the graphs of
y= x^2 and y= 8-x^2. Provide a sketch and use fubini's theorem to determine the order of integration.

2. Determine the best order of integration to find the double integral ??xe^y^2 da Where the region R is in the first quadrant bounded by the graphs of y=x^2,x=0,y=4 provide a sketch of the region.

3.Using Green's theorem evaluate ?(x^2-y^2)dx + (2y-x)dy where c consists of the boundary of the region in the first quadrant that is bounded by the graphs of y=x^2 and y=x^3

4. Using Green's theorem find the work done by the force field;
F= (-16y+sinx^2)i +(4e^y +3x^2)j
Acting along the simple closed curve c
C1- y=x, C2- x^2 + y^2=1, C3- y=-x.

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The solution provides detailed explanation how to find double integrals.

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