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![Page 1: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/1.jpg)
Improper Integrals
Objective: Evaluate integrals that become infinite within the interval
of integration
![Page 2: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/2.jpg)
Improper Integrals
• Our main objective in this section is to extend the concept of a definite integral to allow for infinite intervals of integration and integrands with vertical asymptotes within the interval of integration or one of the bounds.
• We will call the vertical asymptotes infinite discontinuities and we will call integrals with infinite intervals of integration or infinite discontinuities within the interval of integration improper integrals.
![Page 3: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/3.jpg)
Improper Integrals
• Examples of such integrals are:• Infinite intervals of integration.
0
dxex
12x
dx
21 x
dx
![Page 4: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/4.jpg)
Improper Integrals
• Examples of such integrals are:• Infinite discontinuities in the interval of integration.
2
1 1x
dx
3
32x
dx
0
tan xdx
![Page 5: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/5.jpg)
Improper Integrals
• Examples of such integrals are:• Both Infinite discontinuities in the interval of
integration and infinite intervals of integration.
92x
dx
0 x
dx
1
sec xdx
![Page 6: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/6.jpg)
Definition
• The improper integral of f over the interval is defined to be
• In the case where the limit exists, the improper integral is said to converge, and the limit is defined to be the value of the integral. In the case where the limit does not exist, the improper integral is said to diverge, and is not assigned a value.
b
ab
a
dxxfdxxf )(lim)(
),[ a
![Page 7: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/7.jpg)
Example 1a
• Evaluate
12x
dx
![Page 8: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/8.jpg)
Example 1a
• Evaluate
12x
dx
b
b x
dx
x
dx
12
12
lim
![Page 9: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/9.jpg)
Example 1a
• Evaluate
12x
dx
b
b x
dx
x
dx
12
12
lim
b
b
b
b xx
dx
112
1limlim
11
11
b
![Page 10: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/10.jpg)
Example 1b
• Evaluate
1 x
dx
![Page 11: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/11.jpg)
Example 1b
• Evaluate
1 x
dx
b
b x
dx
x
dx
11
lim
![Page 12: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/12.jpg)
Example 1b
• Evaluate
1 x
dx
b
b x
dx
x
dx
11
lim
bb
b
bx
x
dx1
1
lnlimlim
1lnlnb
![Page 13: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/13.jpg)
Example 1c
• Evaluate
13x
dx
![Page 14: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/14.jpg)
Example 1c
• Evaluate
13x
dx
b
b x
dx
x
dx
13
13
lim
![Page 15: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/15.jpg)
Example 1c
• Evaluate
13x
dx
b
b x
dx
x
dx
13
13
lim
b
b
b
b xx
dx
12
13 2
1limlim
2
1
2
1
2
12
b
![Page 16: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/16.jpg)
Improper Integrals
• These examples lead us to this theorem.
if p > 1 if p < 1
divergesp
x
dxp
11
1
![Page 17: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/17.jpg)
Example 1
• On the surface, the graphs of the last three examples seem very much alike and there is nothing to suggest why one of the areas should be infinite and the other two finite. One explanation is that 1/x3 and 1/x2 approach zero more rapidly than 1/x as x approaches infinity so that the area over the interval [1, b] accumulates less rapidly under the curves y = 1/x3 and y = 1/x2 than under y = 1/x. This slight difference is just enough that two areas are finite and one infinite.
![Page 18: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/18.jpg)
Example 3
• Evaluate
0
)1( dxex x
![Page 19: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/19.jpg)
Example 3
• Evaluate
• We need to use integration by parts.
xu 1
0
)1( dxex x
dxdu dxedv xxev
![Page 20: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/20.jpg)
Example 3
• Evaluate
• We need to use integration by parts.
xu 1
0
)1( dxex x
dxdu dxedv xxev
dxexedxex xxx
)1()1(0
![Page 21: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/21.jpg)
Example 3
• Evaluate
• We need to use integration by parts.
xu 1
0
)1( dxex x
dxdu dxedv xxev
dxexedxex xxx
)1()1(0
xxxxx xeexeedxex
0
)1(
![Page 22: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/22.jpg)
Example 3
• Evaluate
0
)1( dxex x
bxb
x xedxex 0
0
lim)1(
![Page 23: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/23.jpg)
Example 3
• Evaluate
0
)1( dxex x
bxb
x xedxex 0
0
lim)1(
bb e
b
lim
![Page 24: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/24.jpg)
Example 3
• Evaluate
• This is of the form so we will use L’Hopital’s Rule
0
)1( dxex x
bb e
b
lim
![Page 25: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/25.jpg)
Example 3
• Evaluate
• We can interpret this to mean that the net signed area between the graph of and the interval is 0.
0
)1( dxex x
01
lim bb e
xexy )1(),0[
bb e
b
lim
![Page 26: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/26.jpg)
Definition 8.8.3
• The improper integral of f over the interval is defined to be
• The integral is said to converge if the limit exists and diverge if it does not.
b
aa
b
dxxfdxxf )(lim)(
],( b
![Page 27: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/27.jpg)
Definition 8.8.3
• The improper integral of f over the interval is defined to be
where c is any real number (we will usually choose 0 to make it easier). The improper integral is said to converge if both terms converge and diverge if either term diverges.
c
c
dxxfdxxfdxxf )()()(
),(
![Page 28: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/28.jpg)
Example 4
• Evaluate
21 x
dx
![Page 29: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/29.jpg)
Example 4
• Evaluate
21 x
dx
2
)(tanlimtanlim1
lim1
10
1
02
02
bxx
dx
x
dxb
b
b
b
b
![Page 30: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/30.jpg)
Example 4
• Evaluate
21 x
dx
2
)(tanlimtanlim1
lim1
10
1
02
02
bxx
dx
x
dxb
b
b
b
b
2
)tan(limtanlim1
lim1
1010
2
0
2
axx
dx
x
dxa
aa
aa
![Page 31: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/31.jpg)
Example 4
• Evaluate
21 x
dx
2
)(tanlimtanlim1
lim1
10
1
02
02
bxx
dx
x
dxb
b
b
b
b
2
)tan(limtanlim1
lim1
1010
2
0
2
axx
dx
x
dxa
aa
aa
22
![Page 32: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/32.jpg)
Definition 8.8.4
• If f is continuous on the interval [a, b], except for an infinite discontinuity at b, then the improper integral of f over the interval [a, b] is defined as
• In the case where the limit exists, the improper integral is said to converge, and the limit is defined to be the value of the integral. If the limit does not exist, the integral is said to diverge.
k
abk
b
a
dxxfdxxf )(lim)(
![Page 33: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/33.jpg)
Definition 8.8.5
• If f is continuous on the interval [a, b], except for an infinite discontinuity at a, then the improper integral of f over the interval [a, b] is defined as
• In the case where the limit exists, the improper integral is said to converge, and the limit is defined to be the value of the integral. If the limit does not exist, the integral is said to diverge.
b
kak
b
a
dxxfdxxf )(lim)(
![Page 34: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/34.jpg)
Definition 8.8.5
• If f is continuous on the interval [a, b], except for an infinite discontinuity at a point c in (a, b), then the improper integral of f over the interval [a, b] is defined as
• The improper integral is said to converge if both terms converge and diverge if either term diverges.
b
c
c
a
b
a
dxxfdxxfdxxf )()()(
![Page 35: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/35.jpg)
Example 6a
• Evaluate
2
1 1 x
dx
![Page 36: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/36.jpg)
Example 6a
• Evaluate
2
1 1 x
dx
2
1
2
1 1lim
1 kk x
dx
x
dx
![Page 37: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/37.jpg)
Example 6a
• Evaluate
2
1 1 x
dx
21
2
1
2
1
|1|lnlim1
lim1 k
kk
kx
x
dx
x
dx
![Page 38: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/38.jpg)
Example 6a
• Evaluate
2
1 1 x
dx
21
2
1
2
1
|1|lnlim1
lim1 k
kk
kx
x
dx
x
dx
|1|ln|1|ln|1|lnlim 2
1kx k
k
![Page 39: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/39.jpg)
Example 6b
• Evaluate
4
13/2)2(x
dx
![Page 40: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/40.jpg)
Example 6b
• Evaluate
4
13/2)2(x
dx
4
23/2
2
13/2
4
13/2 )2()2()2( x
dx
x
dx
x
dx
![Page 41: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/41.jpg)
Example 6b
• Evaluate
4
13/2)2(x
dx
4
23/2
2
13/2
4
13/2 )2()2()2( x
dx
x
dx
x
dx
3/12
13/22
2
13/2
)2(3lim)2(
lim)2(
x
x
dx
x
dxk
k
k
![Page 42: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/42.jpg)
Example 6b
• Evaluate
4
13/2)2(x
dx
4
23/2
2
13/2
4
13/2 )2()2()2( x
dx
x
dx
x
dx
3/12
13/22
2
13/2
)2(3lim)2(
lim)2(
x
x
dx
x
dxk
k
k
3)21(3)2(3lim 3/13/1
2
k
k
![Page 43: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/43.jpg)
Example 6b
• Evaluate
4
13/2)2(x
dx
4
23/2
2
13/2
4
13/2 )2()2()2( x
dx
x
dx
x
dx
3/12
4
3/22
4
23/2
)2(3lim)2(
lim)2(
x
x
dx
x
dxk
kk
![Page 44: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/44.jpg)
Example 6b
• Evaluate
4
13/2)2(x
dx
4
23/2
2
13/2
4
13/2 )2()2()2( x
dx
x
dx
x
dx
3/12
4
3/22
4
23/2
)2(3lim)2(
lim)2(
x
x
dx
x
dxk
kk
33/13/1
223)2(3)24(3lim
k
k
![Page 45: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/45.jpg)
Example 6b
• Evaluate
4
13/2)2(x
dx 3 233
![Page 46: Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration.](https://reader036.fdocuments.us/reader036/viewer/2022081503/56649f555503460f94c78d1c/html5/thumbnails/46.jpg)
Homework
• Page 576
• 1-17 odd
• Section 7.8
• 1-21 odd