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Numerical Integration of Improper Integrals

This document discusses numerical integration techniques for approximating improper integrals of the form ∫∞f(x)dx. It explains that these integrals can be evaluated by considering the limit as t approaches infinity of ∫tf(x)dx. It also notes that for an improper integral to converge, the sum of ∫ba f(x)dx and ∫∞b f(x)dx must converge, and the "tail" term ∫∞b f(x)dx must approach 0 as b approaches infinity. The document proposes using numerical integration to approximate the original improper integral by finding the value of b such that the tail term is small enough to achieve a desired level of accuracy.

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Hashem EL-MaRimey
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192 views2 pages

Numerical Integration of Improper Integrals

This document discusses numerical integration techniques for approximating improper integrals of the form ∫∞f(x)dx. It explains that these integrals can be evaluated by considering the limit as t approaches infinity of ∫tf(x)dx. It also notes that for an improper integral to converge, the sum of ∫ba f(x)dx and ∫∞b f(x)dx must converge, and the "tail" term ∫∞b f(x)dx must approach 0 as b approaches infinity. The document proposes using numerical integration to approximate the original improper integral by finding the value of b such that the tail term is small enough to achieve a desired level of accuracy.

Uploaded by

Hashem EL-MaRimey
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Numerical Integration of Improper Integrals

MA132 Extra Credit Project 1

We consider numerical integration techniques on improper integrals of the form


Z ∞
f (x)dx.
a

Recall from Calculus, integrals of this form are evaluated by considering


Z t
lim f (x)dx,
t→∞ a

and so one must integrate and then evaluate the limit to determine if the integral converges or
diverges. Given that an improper integral converges, we must have that
Z b Z ∞
f (x)dx + f (x)dx
a b

converges and that Z ∞


f (x)dx → 0 as b → ∞.
b
This term is called the tail of the integral.

Idea: Suppose we need numerical integration to appraoximate our original improper integral and
we want to bound our error by some small number ² > 0. Our problem comes down to determining
b so that the tail of the integral is small enough so that the entire integral can be approximated to
our desired accuracy. The idea is to find b so that
Z ∞ ¯Z Z b ¯
¯ ∞ ¯
¯ ¯
f (x)dx < ²/2 ⇒ ¯ f (x)dx − f (x)dx¯ < ²/2.
b ¯ a a ¯
Rb
So if we approximate I ≈ a f (x)dx to an accuracy of ²/2 then have that
¯Z ∞ ¯
¯ ¯
¯ f (x) − I ¯
¯ ¯
a
¯Z Z ∞ ¯
¯ b ¯
¯ ¯
= ¯ f (x)dx + f (x)dx − I ¯
¯ a b ¯
¯Z ¯ ¯Z ¯
¯ b ¯ ¯ ∞ ¯
¯ ¯ ¯
≤ ¯ f (x)dx − I ¯ + ¯ f (x)dx¯¯
¯ a ¯ b

≤ ²/2 + ²/2 = ²
Question:
What is the error associated with composite Simspon’s rule? Use this to determine how many
intervals are needed to compute Z ∞
1 −x
e dx
1 x
with composite Simpson’s rule so that the error is less than 10−6 . Your final product should be a
thorough description of this solution process, using complete sentences and proper English. You
do not need to type this up, but your work should be neat, organized, and fully explained.
HINTS:

1. You need to find b so the tail of the integral is small enough.


R ∞ −x
2. First consider b e dx. Does this improper integral converge?

3. You will need the fact that


Z ∞ Z ∞
1 −x
e dx ≤ e−x dx . . . Why??
1 x 1

4. Once you
Rb
know b you need to use the error term for Simpson’s rule to determine what n is
for I ≈ a f (x)dx < ²/2

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