Iterated Integrals

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Section 16.2

• Iterated Integrals

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PARTIAL INTEGRATION
We use the notation to
mean that x is held fixed and that f (x, y) is
integrated with respect to y from y c to y d.
This procedure is called partial integration
with respect to y. There is an analogous
definition of partial integration with respect to
x.
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ITERATED INTEGRALS
Note that is
a function that depends on x only. If we now
integrate the function A with respect to x from x
a to x b, we get. This equation is called an
iterated integral.
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MORE ON ITERATED INTEGRALS

means the we first integrate
with respect to y from c to d and then with
respect to x from a to b.

means the we first integrate
with respect to x from a to b and then with
respect to y from c to d.
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FUBINIS THEOREM
Theorem If f is continuous on the rectangle
R  (x, y) a x b, c y d
then More generally, this is true if we assume
that f is bounded on R, f is discontinuous
only on a finite number smooth curves, and the
iterated integrals exist.
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A SPECIAL CASE OF DOUBLE INTEGRALS
Suppose the function f (x, y) can be factored
as the product of a function of x only and a
function of y only that is, f (x, y) g(x)
h(y). The double integral can be written as the
product of two integrals.