Section 7.7: Improper Integrals
3
1
If
1 x 2 dx makes sense, then what about

Definition


1
1
dx ?
2
x
N
 f ( x) dx  lim  f ( x) dx
1
N 
1
This is our first example of an improper integral.

1
1 x 2 dx

N
 lim
N 
1
1 x dx  Nlim

N

1
1
 1 1 
dx  lim   
2
N  N
1 
x

 1
1
ln | N |  ln |1|  
1 x dx  Nlim

If the limit is finite then we say that
the improper integral converges
otherwise the integral diverges.
Notice that if the integral is going to converge,
Zero must be a horizontal asymptote
and the function must get small quickly.

Fact
1
1 x p dx converges  p  1


1
1
dx
x
 lim 2 1  2 N  
N 
Vertical Asymptotes provide another kind of improper integral.
You always have to check that the function is continuous.
3
N
3
1
dx
dx
dx
 lim 
0 x  1  Nlim
 
1
N 1
x 1
x 1
0
N
lim ln | N  1|  ln | 0  1|
N 1
lim ln | N  1|  ln | 3  1|
N 1
Diverges, but if we ignore the asymptote, we get the wrong answer:
ln | x  1|30  ln 2
Third Type

1
 1  x 2 dx 
0
1
 1  x 2 dx 

1
0 1  x 2 dx
lim tan 1 (0)  tan 1 ( N )  lim tan 1 ( N )  tan 1 (0)
N 
N 
 

 

 0
 0 
  
2 

 2


Gabriel’s horn
Rotate f(x) = 1/x about x-axis

2
1
Volume =     dx  
x
1

1
1 


Surface Area = 2   1  
1  x   x 2  dx
2
Diverges!!
This is a paint can that can be filled with π unit3 of paint,
but the surface requires an infinite amount of paint!