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Integral
calculus |
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The improper integrals
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The improper integral definition
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The improper integrals examples
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The
improper integrals
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A
definite integral with one or both limits of integration infinite,
or having an integrand that becomes infinite between
the limits of integration is called the improper
integral.
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Let
f (x)
be defined on [a,
oo
)
and integrable on [a,
b] for all a
< b <
oo,
then
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provided
the limit exists.
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If
f (x)
is defined on ( -
oo,
b] and integrable on [a,
b] for all -
oo
< a < b,
then
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provided
the limit exists.
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If
f (x)
is defined on ( -
oo,
oo
)
and integrable on any closed interval [a,
b], then
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provided
the limits on the right exist.
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In
each case where the appropriate limit exists we say the integral
converges, otherwise the integral is said to diverge
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Example:
Evaluate |
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As
the area of the region beneath the graph of f
over the interval [1,
b] as b
goes to infinity A
= 1, thus the
integral
converges.
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Example:
Evaluate |
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Therefore,
the integral diverges.
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Example:
Evaluate |
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Therefore,
the integral converges.
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Example:
Evaluate |
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Therefore,
the integral converges.
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Example:
Evaluate |
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Therefore,
the integral converges.
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Example:
Evaluate |
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Therefore,
the integral converges.
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Functions
contents G |
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