Iterated Integrals
You can enter and evaluate iterated integrals
. If a≤b,
f (x)≤g(x) for all
x∈
a, b
, and
k(x, y)≥ 0 for all
x∈
a, b
and all
y∈
f (x), g(x)
then the iterated integral

k(
x,
y) d
y d
x
can be interpreted as the volume of the solid bounded by
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a≤x≤b |
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f (x)≤y≤g(x) |
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0≤z≤k(x, y) |
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Here are a few examples of iterated integrals.
bfIntegral sfEvaluate sfEvaluate Numerically
6pt x2cos y dy dx
= cos 1 + 2 sin 1 - 2
= .2232442755
 ex2 dx dy 6pt
= e9 -
= 1350.347321
 sin x2 dy dx 6pt
= - cos 1 +
= .2298488471
  dy dx
= - +4 ln 2
= .2725887222 |
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Other multiple integrals look innocent enough
but require a bit more effort to evaluate. Attempting to evaluate the double
integral
exactly leads to frustration. However, you can reverse the order of
integration by looking carefully at the region of integration in the plane.
dtbpF3in2.0003in0ptThis region is bounded above
by y = x2 and bounded below by y = 0. The new integral is
This double integral can be solved by iterated integration. The inner
integral is just
You can integrate the resulting outer integral

x2 dx by applying Calculus + Change Variable,
say with
u2 = x3 + 1. Then choose Evaluate and Evaluate
Numerically, to get
 x2 dx |
= |
 u2 du |
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= |
 -  |
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= |
.4063171388 |
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