Double integrals First, let's recall Riemann sums for integrals - PDF document

Jul 22, 2023 •18 likes •208 views

Double integrals First, let's recall Riemann sums for integrals with a single variable... 1 2 3 4 5 Some comments: (1) Note that we can estimate double integrals using double Riemann sums. While this is tedious by hand, computers do this

Double integrals First, let's recall Riemann sums for integrals with a single variable... 1
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Some comments: (1) Note that we can estimate double integrals using double Riemann sums. While this is tedious by hand, computers do this type of thing routinely. (2) The interpretation as volume is only for surfaces f(x,y)>0; otherwise we get regions with "negative" volume. Volume calculation is not the motivation anyway, just a way to start thinking about the Riemann sum. 6
Midpoint rule: Use the centers of rectangles as sample points in the double Riemann sum. 7
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Average value 9
Calculation of integrals: iterated integrals. 10
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Fubini's Theorem *So you can switch the order of integration for convenience... very useful sometimes. *Here f(x,y) can be discontinuous on a finite set of curves so long as it is also bounded and the iterated integrals exist. 13
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Practice! 16
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