Improper Integrals
MATH 211, Calculus II
J. Robert Buchanan
Department of Mathematics
Summer 2008
J. Robert Buchanan Improper Integrals
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Improper Integrals: Definition, Convergence, and Examples, Exams of Calculus
This document from math 211, calculus ii, covers improper integrals, their definition, and examples of their convergence or divergence. How to evaluate definite integrals with isolated discontinuities and improper integrals over infinite intervals. It also includes the comparison test and graphical approach for determining convergence.
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Pre 2010
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Improper Integrals
MATH 211, Calculus II
J. Robert Buchanan
Department of Mathematics
Summer 2008
Definite Integrals
Theorem (Fundamental Theorem of Calculus (Part I)) If f is continuous on [a, b] then ∫ (^) b
a
f (x) dx = F (b) − F (a)
where F is any antiderivative of f on (a, b).
Question: Can we evaluate the definite integral
− 1
x^2
dx?
Improper Integrals
Extra care must be exercised when attempting to evaluate definite integrals for which the integrand possesses isolated discontinuities within the integration interval,
and/or
the interval over which we integrate is of infinite length.
Improper Integrals
Extra care must be exercised when attempting to evaluate definite integrals for which the integrand possesses isolated discontinuities within the integration interval,
and/or
the interval over which we integrate is of infinite length.
Examples
Example Determine if the following improper integrals converge or diverge.
1
1
2 x x^2 − 1
dx
2
0
√ (^3) x dx
3
∫ (^) π/ 2
0
tan x dx
Discontinuity in (a, b)
Definition Suppose f is continuous on [a, b] except at some c ∈ (a, b) and |f (x)| → ∞ as x → c. The improper integral is ∫ (^) b
a
f (x) dx =
∫ (^) c
a
f (x) dx +
∫ (^) b
c
f (x) dx.
If
∫ (^) c
a
f (x) dx = L 1 and
∫ (^) b
c
f (x) dx = L 2 the improper integral ∫ (^) b
a
f (x) dx converges to L 1 + L 2. If either of the improper
integrals
∫ (^) c
a
f (x) dx or
∫ (^) b
c
f (x) dx diverges then
∫ (^) b
a
f (x) dx diverges as well.
Second Type
Definition If f is continuous on [a, ∞) we define the improper integral ∫ (^) ∞
a
f (x) dx = lim R→∞
∫ R
a
f (x) dx.
If f is continuous on (−∞, a] we define the improper integral ∫ (^) a
−∞
f (x) dx = lim R→−∞
∫ (^) a
R
f (x) dx.
If the limit is L (finite) we say the improper integral converges , otherwise we say it diverges.
Examples
Example Determine if the following improper integrals converge or diverge. 1
1
e−x^ dx
2
5
x
dx
3
5
x^2
dx
Example
Example Determine if the following improper integral converges. ∫ (^) ∞
−∞
ex^ + e−x^
dx
Graphical Approach
Suppose f and g are two continuous functions defined on [a, ∞) and such that 0 ≤ f (x) ≤ g(x) for all x ≥ a.
a x
y
fHxL gHxL
a x
y
If
a
f (x) dx diverges, what about
a
g(x) dx?
If
a
g(x) dx converges, what about
a
f (x) dx?
Graphical Approach
Suppose f and g are two continuous functions defined on [a, ∞) and such that 0 ≤ f (x) ≤ g(x) for all x ≥ a.
a x
y
fHxL gHxL
a x
y
If
a
f (x) dx diverges, what about
a
g(x) dx?
If
a
g(x) dx converges, what about
a
f (x) dx?
Comparison Test
Theorem (Comparison Test) Suppose that f and g are continuous on [a, ∞) and 0 ≤ f (x) ≤ g(x) for all x ≥ a.
(^1) If
a
g(x) dx converges, then
a
f (x) dx converges.
(^2) If
a
f (x) dx diverges, then
a
g(x) dx diverges.
Homework
Read Section 6.6 and work exercises 1–59 odd.