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Improp= er Integrals
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Improp= er Integrals
Integ= rals over an infinite interval such as ∫ e-= x dx =
0
∞
------=_NextPart_01C97CE5.C5E209D0 Content-Location: file:///C:/D0AA91F3/21ImproperIntegrals_files/slide0009.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Improp= er Integrals
Integrals over an infinite interval such as ∫ e-x dx        = ;            &n= bsp;            = ;    
        = ;            &n= bsp;            = ;     or        = ;            &n= bsp;        
0
∞
integrals of f(x) that goes to ∞ within the interval of
integration such as ∫ Ln(x) dx,
0
1
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Improp= er Integrals
Integrals over an infinite interval such as ∫ e-x dx        = ;            &n= bsp;            = ;    
        = ;            &n= bsp;            = ;     or        = ;            &n= bsp;        
0
∞
integrals of f(x) that goes to ∞ within the interval of
integration such as ∫ Ln(x) dx, are called &= #13;
improper integrals.        = ;            &n= bsp;         
0
1
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Improp= er Integrals
Integrals over an infinite interval such as ∫ e-x dx        = ;            &n= bsp;            = ;    
        = ;            &n= bsp;            = ;     or        = ;            &n= bsp;        
0
∞
integrals of f(x) that goes to ∞ within the interval of
integration such as ∫ Ln(x) dx, are called &= #13;
improper integrals.        = ;            &n= bsp;         
0
1
Let F(= x) be the antiderivative of f(x).  =
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Improp= er Integrals
Integrals over an infinite interval such as ∫ e-x dx        = ;            &n= bsp;            = ;    
        = ;            &n= bsp;            = ;     or        = ;            &n= bsp;        
0
∞
integrals of f(x) that goes to ∞ within the interval of
integration such as ∫ Ln(x) dx, are called &= #13;
improper integrals.        = ;            &n= bsp;         
0
1
Let F(= x) be the antiderivative of f(x).  = We define        = ;            &n= bsp;        
∫
a
∞
f(x) d= x as lim    f(x) dx        = ;         
∫
a
u
u à&= #8734;
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Improp= er Integrals
Integrals over an infinite interval such as ∫ e-x dx        = ;            &n= bsp;            = ;    
        = ;            &n= bsp;            = ;     or        = ;            &n= bsp;        
0
∞
integrals of f(x) that goes to ∞ within the interval of
integration such as ∫ Ln(x) dx, are called &= #13;
improper integrals.        = ;            &n= bsp;         
0
1
Let F(= x) be the antiderivative of f(x).  = We define        = ;            &n= bsp;        
∫
a
∞
f(x) d= x as lim    f(x) dx =3D = F(x)| =3D lim F(x) – F(a)        = ;        
∫
a
u
u à&= #8734;
x à&= #8734;
a
∞
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Improp= er Integrals
Integrals over an infinite interval such as ∫ e-x dx        = ;            &n= bsp;            = ;    
        = ;            &n= bsp;            = ;     or        = ;            &n= bsp;        
0
∞
integrals of f(x) that goes to ∞ within the interval of
integration such as ∫ Ln(x) dx, are called &= #13;
improper integrals.        = ;            &n= bsp;         
0
1
Let F(= x) be the antiderivative of f(x).  = We define        = ;            &n= bsp;        
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
&#= 8734;
e-x dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
a
∞
f(x) d= x as lim    f(x) dx =3D = F(x)| =3D lim F(x) – F(a)        = ;        
∫
a
u
u à&= #8734;
x à&= #8734;
a
∞
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Improp= er Integrals
Integrals over an infinite interval such as ∫ e-x dx        = ;            &n= bsp;            = ;    
        = ;            &n= bsp;            = ;     or        = ;            &n= bsp;        
0
∞
integrals of f(x) that goes to ∞ within the interval of
integration such as ∫ Ln(x) dx, are called &= #13;
improper integrals.        = ;            &n= bsp;         
0
1
Let F(= x) be the antiderivative of f(x).  = We define        = ;            &n= bsp;        
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
&#= 8734;
e-x dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
&#= 8734;
e-x dx =3D -e-x|        = ;            &n= bsp;            = ;            &n= bsp;        
0
&#= 8734;
∫
a
∞
f(x) d= x as lim    f(x) dx =3D = F(x)| =3D lim F(x) – F(a)        = ;        
∫
a
u
u à&= #8734;
x à&= #8734;
a
∞
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Improp= er Integrals
Integrals over an infinite interval such as ∫ e-x dx        = ;            &n= bsp;            = ;    
        = ;            &n= bsp;            = ;     or        = ;            &n= bsp;        
0
∞
integrals of f(x) that goes to ∞ within the interval of
integration such as ∫ Ln(x) dx, are called &= #13;
improper integrals.        = ;            &n= bsp;         
0
1
Let F(= x) be the antiderivative of f(x).  = We define        = ;            &n= bsp;        
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
&#= 8734;
e-x dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
&#= 8734;
e-x dx =3D -e-x| =3D lim (-e-x) – (-e0)        = ;            &n= bsp;            = ;         
x à∞
0
&#= 8734;
∫
a
∞
f(x) d= x as lim    f(x) dx =3D = F(x)| =3D lim F(x) – F(a)        = ;        
∫
a
u
u à&= #8734;
x à&= #8734;
a
∞
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Improp= er Integrals
Integrals over an infinite interval such as ∫ e-x dx        = ;            &n= bsp;            = ;    
        = ;            &n= bsp;            = ;     or        = ;            &n= bsp;        
0
∞
integrals of f(x) that goes to ∞ within the interval of
integration such as ∫ Ln(x) dx, are called &= #13;
improper integrals.        = ;            &n= bsp;         
0
1
Let F(= x) be the antiderivative of f(x).  = We define        = ;            &n= bsp;        
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
&#= 8734;
e-x dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
&#= 8734;
e-x dx =3D -e-x| =3D lim (-e-x) – (-e0) =3D 0 – = (-1) =3D 1        = ;            &n= bsp;            = ;            &n= bsp;       
x à∞
0
&#= 8734;
∫
a
∞
f(x) d= x as lim    f(x) dx =3D = F(x)| =3D lim F(x) – F(a)        = ;        
∫
a
u
u à&= #8734;
x à&= #8734;
a
∞
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Improp= er Integrals
Let F(x) be the antiderivative of f(x) and lim f(= x) &agrav= e; ∞.
        = ;            &n= bsp;         
x àa
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Improp= er Integrals
Let F(x) be the antiderivative of f(x) and lim f(= x) &agrav= e; ∞.
We define        = ;            &n= bsp;         
x àa
∫
a
b
f(x) = dx =3D F(x)| =3D F(b) – limF(x)        = ;        
x àa
a
b
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Improp= er Integrals
Let F(x) be the antiderivative of f(x) and lim f(= x) &agrav= e; ∞.
We define        = ;            &n= bsp;         
x àa
∫
a
b
f(x) = dx =3D F(x)| =3D F(b) – limF(x)        = ;        
x àa
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-1/2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
a
b
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Improp= er Integrals
Let F(x) be the antiderivative of f(x) and lim f(= x) &agrav= e; ∞.
We define        = ;            &n= bsp;         
x àa
∫
a
b
f(x) = dx =3D F(x)| =3D F(b) – limF(x)        = ;        
x àa
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-1/2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-1/2 dx =3D 2x1/2 |        = ;            &n= bsp;            = ;            &n= bsp;        
1<= /b>
0
a
b
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Improp= er Integrals
Let F(x) be the antiderivative of f(x) and lim f(= x) &agrav= e; ∞.
We define        = ;            &n= bsp;         
x àa
∫
a
b
f(x) = dx =3D F(x)| =3D F(b) – limF(x)        = ;        
x àa
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-1/2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-1/2 dx =3D 2x1/2 |&= nbsp; =3D 2 – lim(2x1/2)        = ;            &n= bsp;            = ;            &n= bsp;        
1<= /b>
0
x à0
a
b
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Improp= er Integrals
Let F(x) be the antiderivative of f(x) and lim f(= x) &agrav= e; ∞.
We define        = ;            &n= bsp;         
x àa
∫
a
b
f(x) = dx =3D F(x)| =3D F(b) – limF(x)        = ;        
x àa
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-1/2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-1/2 dx =3D 2x1/2 |&= nbsp; =3D 2 – lim(2x1/2) =3D 2        = ;            &n= bsp;            = ;            &n= bsp;       
1<= /b>
0
x à0
a
b
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Improp= er Integrals
Let F(x) be the antiderivative of f(x) and lim f(= x) &agrav= e; ∞.
We define        = ;            &n= bsp;         
x àa
∫
a
b
f(x) = dx =3D F(x)| =3D F(b) – limF(x)        = ;        
x àa
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-1/2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-1/2 dx =3D 2x1/2 |&= nbsp; =3D 2 – lim(2x1/2) =3D 2        = ;            &n= bsp;            = ;            &n= bsp;       
1<= /b>
0
x à0
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
a
b
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Improp= er Integrals
Let F(x) be the antiderivative of f(x) and lim f(= x) &agrav= e; ∞.
We define        = ;            &n= bsp;         
x àa
∫
a
b
f(x) = dx =3D F(x)| =3D F(b) – limF(x)        = ;        
x àa
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-1/2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-1/2 dx =3D 2x1/2 |&= nbsp; =3D 2 – lim(2x1/2) =3D 2        = ;            &n= bsp;            = ;            &n= bsp;       
1<= /b>
0
x à0
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-2 dx =3D –x–1 |        = ;            &n= bsp;            = ;            &n= bsp;        
1<= /b>
0
a
b
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Improp= er Integrals
Let F(x) be the antiderivative of f(x) and lim f(= x) &agrav= e; ∞.
We define        = ;            &n= bsp;         
x àa
∫
a
b
f(x) = dx =3D F(x)| =3D F(b) – limF(x)        = ;        
x àa
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-1/2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-1/2 dx =3D 2x1/2 |&= nbsp; =3D 2 – lim(2x1/2) =3D 2        = ;            &n= bsp;            = ;            &n= bsp;       
1<= /b>
0
x à0
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-2 dx =3D –x–1 |  =3D -1 – lim(-x-1)        = ;            &n= bsp;            = ;            &n= bsp;        
1<= /b>
0
x à0
a
b
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Improp= er Integrals
Let F(x) be the antiderivative of f(x) and lim f(= x) &agrav= e; ∞.
We define        = ;            &n= bsp;         
x àa
∫
a
b
f(x) = dx =3D F(x)| =3D F(b) – limF(x)        = ;        
x àa
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-1/2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-1/2 dx =3D 2x1/2 |&= nbsp; =3D 2 – lim(2x1/2) =3D 2        = ;            &n= bsp;            = ;            &n= bsp;       
1<= /b>
0
x à0
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-2 dx =3D –x–1 |  =3D -1 – lim(-x-1) =3D -1 –= ; (-∞) =3D W= 34;        = ;            &n= bsp;            = ;            &n= bsp;       
1<= /b>
0
x à0
a
b
------=_NextPart_01C97CE5.C5E209D0 Content-Location: file:///C:/D0AA91F3/21ImproperIntegrals_files/slide0028.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Improp= er Integrals
Let F(x) be the antiderivative of f(x) and lim f(= x) &agrav= e; ∞.
We define        = ;            &n= bsp;         
x àa
∫
a
b
f(x) = dx =3D F(x)| =3D F(b) – limF(x)        = ;        
x àa
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-1/2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-1/2 dx =3D 2x1/2 |&= nbsp; =3D 2 – lim(2x1/2) =3D 2        = ;            &n= bsp;            = ;            &n= bsp;       
1<= /b>
0
x à0
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-2 dx =3D –x–1 |  =3D -1 – lim(-x-1) =3D -1 –= ; (-∞) =3D W= 34;        = ;            &n= bsp;            = ;            &n= bsp;       
1<= /b>
0
x à0
If the improper integral exists, we say it converges.        = ;            &n= bsp;         
a
b
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Improp= er Integrals
Let F(x) be the antiderivative of f(x) and lim f(= x) &agrav= e; ∞.
We define        = ;            &n= bsp;         
x àa
∫
a
b
f(x) = dx =3D F(x)| =3D F(b) – limF(x)        = ;        
x àa
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-1/2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-1/2 dx =3D 2x1/2 |&= nbsp; =3D 2 – lim(2x1/2) =3D 2        = ;            &n= bsp;            = ;            &n= bsp;       
1<= /b>
0
x à0
Example:  Find        = ;            &n= bsp;            = ;            &n= bsp;          
∫
0
1<= /b>
x-2 dx        = ;            &n= bsp;            = ;            &n= bsp;         
∫
0
1<= /b>
x-2 dx =3D –x–1 |  =3D -1 – lim(-x-1) =3D -1 –= ; (-∞) =3D W= 34;        = ;            &n= bsp;            = ;            &n= bsp;       
1<= /b>
0
x à0
If the improper integral exists, we say it converges.        = ;            &n= bsp;         
If the improper integral fails to exist or its infinite, we = say it diverges.        = ;            &n= bsp;         
a
b
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Improp= er Integrals
(The Fl= oor theorem)
------=_NextPart_01C97CE5.C5E209D0 Content-Location: file:///C:/D0AA91F3/21ImproperIntegrals_files/slide0033.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Improp= er Integrals
(The Floor theorem)
y =3D f(x)
y =3D g(= x)
∞=
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Improp= er Integrals
(The Floor theorem)
If f(x) > g(x) > 0 and    g(x) dx =3D ∞,
∫
a
b
y =3D f(x)
y =3D g(= x)
∞=
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Improp= er Integrals
(The Floor theorem)
If f(x) > g(x) > 0 and    g(x) dx =3D ∞, then    f(x) =3D ∞.
∫
a
b