Math 152 – Spring 2016
Section 8.9
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Section 8.9 – Improper Integration
Example 1. Find the area under f (x) =
1
x2
between 1 and t.
Definition: Improper Integral, Type 1
• If
Rt
a
f (x) dx exists for every number t ≥ a, then
∞
Z
t
Z
f (x) dx
f (x) dx = lim
t→∞
a
• If
Rb
t
a
f (x) dx exists for every number t ≤ b, then
Z
b
Z
f (x) dx = lim
t→−∞
−∞
b
f (x) dx
t
The improper integrals above are called convergent if the limit exists and divergent if the limit does not exist.
R∞
Ra
• If both a f (x) dx and −∞ f (x) dx are convergent, then we define
Z
∞
Z
a
f (x) dx =
−∞
Z
f (x) dx +
−∞
∞
f (x) dx
a
Example 2. Determine whether the following converge or diverge. If the integral
converges, evaluate it.
R∞
(a) 1 x1 dx
(b)
R0
−∞
xex dx
Math 152 – Spring 2016
(c)
R∞
1
−∞ 1+x2
Section 8.9
dx
(d) For what values of p is the integral
Theorem.
R∞
1
1
xp
R∞
1
1
xp
dx convergent?
dx is convergent if p > 1 and divergent if p ≤ 1.
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Math 152 – Spring 2016
Section 8.9
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Definition: Improper Integral, Type 2
• If f is continuous on [a, b) and is discontinuous at b, then
b
Z
a
t
Z
f (x) dx = lim−
t→b
f (x) dx
a
if this limit exists (as a finite number).
• If f is continuous on (a, b] and is discontinuous at a, then
Z
a
b
Z
f (x) dx = lim+
t→a
b
f (x) dx
t
The improper integrals above are called convergent if the limit exists and divergent if the limit does not exist.
Rc
Rb
• If f has a discontinuity at c, where a < c < b, and both a f (x) dx and c f (x) dx
are convergent, then we define
Z
b
Z
f (x) dx =
a
c
Z
f (x) dx +
a
b
f (x) dx
c
Example 3. Determine whether the following converge or diverge. If the integral
converges, evaluate it.
R0 1
(a) −2 x+2
(b)
R π/2
0
sec x dx
Math 152 – Spring 2016
(c)
R4
1
0 (x−3)2
Section 8.9
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dx
Suppose you did not notice this was improper. You would do the following which
gives the WRONG ANSWER!
Z
0
4
4
1
1
dx = −
(x − 3)2
x−3 0
1 1
+
=−
4 3
7
=−
12
7
But this integral is DIVERGENT, NOT equal to − 12
.
(d)
R1
0
ln x dx
Math 152 – Spring 2016
Section 8.9
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Comparison Theorem Suppose that f and g are continuous functions with
f (x) ≥ g(x) ≥ 0 for x ≥ a
R∞
R∞
• If a f (x) dx is convergent, then a g(x) dx is convergent.
R∞
R∞
• If a g(x) dx is divergent, then a f (x) dx is divergent.
Example 4. Use the comparison theorem to determine if the following converges or
diverges.
R∞
(a) 1 √xx5 +2 dx
(b)
R π/2
0
1
x cos2 x
dx