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Evaluating a double integral as an iterated integral: Use Fubini’s theorem to evaluate the given double integrals. For each integral, show that you obtain the same result when you integrate using both possible orders of integration when R = {(x, y) | 0 ≤ x ≤ 2 and 1 ≤ y ≤ 4}.

(x+2y)dA

Short Answer

Expert verified

(x+2y)dA=36Squareunits

Step by step solution

01

Fubini’s theorem 

Let a< b and c<d be real numbers, let R be the rectangle defined by,

R=(x,y)|axbandcydand let f(x) is continuous on R, Then

f(x,y)dA=abcdf(x,y)dxdy

02

Evaluate the integral

(x+2y)dA=1402(x+2y)dxdy=1402(x+2y)dxdy=14dyx22+2xy02=14dy222+2×2×y=14(2+4y)dy=2y+4y2214=8+32-2+2=40-4=36

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