Chapter 5: Double Integration
Section 5.2: Iterated Double Integrals
If fx,y is continuous over the rectangle R defined by a≤x≤b,c≤y≤d, its double integral over R can be evaluated by either of the two iterated integrals
∫∫Rfx,y dA = ∫ab∫cdfx,y dy dx = ∫cd∫abfx,y dx dy
There are weaker conditions under which the double integral of f can be evaluated by iteration, but in a first course in multivariate calculus, continuity suffices.
The mechanics of the iterated integral are detailed in the following examples.
If fx,y=x2+y2, and R is the square region 0≤x,y≤1, evaluate ∫∫Rf dA by both possible iterations.
If fx,y=7−3 x2−5 y2, and R is the rectangular region −1≤x≤1,−1/2≤y≤1/2, evaluate ∫∫Rf dA by both possible iterations.
If fx,y=1+5 x2+7 y2, and R is the rectangular region −1≤x≤1,−2≤y≤2, evaluate ∫∫Rf dA by both possible iterations.
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