math%104%–calculus % 8.8%improper%integrals% · 2015. 3. 2. · • an integral can be called...

Math 104 -‐ Yu

Math 104 – Calculus 8.8 Improper Integrals

Math 104 -‐ Yu

Improper Integrals •  Goal: To evaluate integrals of func?ons over infinite intervals or with an infinite discon?nuity.

•  Method: We replace the “bad” endpoints with variables and take limits. Combines skills of integra?on and evalua?ng limits.

•  Descrip2on: If the limit exists, we say the integral converges and if it fails to exist (this includes infinite limits), we say the integral diverges.

Math 104 -‐ Yu

Types of Improper Integrals An integral can be called improper with one or any combina?on of the following: Type I: contains infinity •  at upper limit

•  lower limit

•  or both

Nicolas Fraiman Math 104

Types of improper integrals• An integral can be called improper with one or any

combination of the following

• Type I: Infinite interval   at upper limit, lower limit  or both  

Z 1

1

lnx

x

2dx





Z 0

�1

1

1 + x2dx


• An integral can be called improper with one or any combination of the following


Types of improper integrals

Z 1

�1

1

1 + x2dx

Math 104 -‐ Yu

Types of Improper Integrals Type II Infinite discon?nuity •  at lower limit

•  upper limit

•  so some value in between





• Type II: Infinite discontinuity   at upper limit, lower limit, or  some value in between






Z 1

0

1

(x� 1)2/3dx






Z 3

1

1

(x� 1)2/3dx

Z 3

0

1

(x� 1)2/3dx

Math 104 -‐ Yu

Type I Improper Integral

Both have to converge to ensure that the LHS converge

Math 104 -‐ Yu

Examples


Examples 1. Compute if it exists.     2. Determine whether converges or diverges.      3. Evaluate if it converges.

Z 1

1e

�2xdx

Z 1

2

1px

dx

Z 1

�1

x

2

1 + x6dx



Z 1

1e

�2xdx

Z 1

2

1px

dx

Z 1

�1

x

2

1 + x6dx

Math 104 -‐ Yu

Examples



Z 1

1e

�2xdx

Z 1

2

1px

dx

Z 1

�1

x

2

1 + x6dx

Math 104 -‐ Yu

Type II Integrals


Type II integrals

Both have to converge to ensure that the LHS converge

Math 104 -‐ Yu

Examples


Examples 4. Find      5. Classify      6. Double improper

Z 9

0

1px

dx.

Z 2

�1

1

x

4dx.

Z 1

0

e

�1/x

x

2dx.

5. Determine of what type is the integral

Z 2

�1

1

x

4dx. Does it converge? If

it converges, evaluate the integral.

Math 104 -‐ Yu

Examples In general, we can have mixture of Type I and Type II integrals. 6. Determine the type(s) of the integral

Z 1

0

e

�1/x

x

2dx. Does it converge?

If it converges, evaluate the integral.

Math 104 -‐ Yu

Improper Integral of 1/x^p For any c > 0,

•Z 1

c

1

x

pdx converges for p > 1, diverges for p 1.

•Z c

0

1

x

pdx converges for p < 1, diverges for p � 1.

Math 104 -‐ Yu

Direct Comparison


Comparison theorems


Comparison theorems

WARNING: Do NOT switch the conditionsand the implications.

Math 104 -‐ Yu

Examples


Examples 7. Does converge or diverge?       8. Same for

Z 1

1

2 + sinx

x

2dx

Z 4

0

x+ 3

x

2dx

8. Does

Z 4

0

x+ 3

x

2dx converge or diverge?

Math 104 -‐ Yu

Limit Comparison


Comparison theorems


Comparison theorems

i.e., f ⇠ g, f is ofthe same order as g

Math 104 -‐ Yu

Examples


Examples 9. Does converge or diverge?        10. A harder example

Z 1

2

xpx

4 + 5dx

Z ⇡/2

0cot(x) dx.

Math 104 -‐ Yu

Comparison Tests for Type II •  Comparison Tests also work for improper integral of the second type by replacing in the limit with the point where the func?on goes to infinity.

1

10. Does

Z ⇡/2

0cot(x)dx converge or diverge?

Math 104 -‐ Yu

p-‐test for type I integral •  In many cases, we can compare the integrand f(x) with func?ons of the form 1/x^p.

Type I integral:

Z 1

cf(x)dx, where f(x) is a nonnegative function and

c > 0.

• If f(x) ⇠ 1x

pwhen x ! 1, then the integral converges if p > 1,

diverges if p 1.

• If f(x) Cx

pfor some constant C > 0 and p > 1, when x is large

enough, then the integral converges.

• If f(x) � Cx

pfor some constant C > 0 and p < 1, when x is large

enough, then the integral diverges.

Math 104 -‐ Yu

p-‐test for type II integral Type II integral:

Zb

a

f(x)dx, where f(x) is a nonnegative function con-

tinuous over the interval (a, b] and lim

x!a+f(x) = 1,

• If f(x) ⇠ 1(x� a)p when x ! a

+,

i.e., lim

x!a+f(x)

1(x�a)p

= L where L is a finite positive number,

then the integral converges if p < 1, diverges if p � 1.

• If f(x) C(x� a)p for some constant C > 0 and p < 1, when x is

close to a, then the integral converges.

• If f(x) � C(x� a)p for some constant C > 0 and p � 1, when x is

close to a, then the integral diverges.

math%104%–calculus % 8.8%improper%integrals% · 2015. 3. 2. · • an integral can be called...

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