1. CHAPTER 2 Improper Integrals 2.4 Continuity 1. Infinite Intervals
Discontinuous functions

2. CHAPTER 2 Improper Integrals 2.4 Continuity 1. Infinite Intervals
a. If af (x) dx exists for every number t  a, then
af (x) dx = lim t ->  at f (x) dx
provided this limit exists as a finite number.
animation

3. CHAPTER 2
The improper integrals af (x) dx and -b f (x) dx
are called convergent if the corresponding limit exists
And divergent if the limit does not exist.
2.4 Continuity
1  (1 / x p) dx is convergent if p > 1 and divergent if p  1.

4. CHAPTER 2 2.4 Continuity -  f (x) dx = -a f (x) dx + af (x) dx
c. If both af (x) dx and -b f (x) dx are convergent,
then we define number t  b, then
-  f (x) dx = -a f (x) dx + af (x) dx
( any each number a can be used).
2.4 Continuity

5. Example: Determine whether the integral -0 e3xdt is divergent or convergent.
Example: Determine whether the integral - ( 1/(x + 3)) dx is divergent or convergent.
Example: Is the integral 12 ln (x - 1) dx improper? Why?

6. CHAPTER 2 2.4 Continuity Discontinuous functions:
If f is continuous on [a,b) and is discontinuous at b, then abf (x) dx = lim t -> b- at f (x) dx if this limit exists (as a finite number).
animation

7. CHAPTER 2 2.4 Continuity Discontinuous functions:
b. If f is continuous on (a,b] and is discontinuous at a, then abf (x) dx = lim t -> a+ bt f (x) dx if this limit exists (as a finite number).
The improper integral abf (x) dx is called convergent
if the corresponding limit exists and divergent if the
limit does not exist.

8. If f has discontinuity at c, where a < c < b, and both acf (x) dx and cb f (x) dx are convergent, then we define
abf (x) dx = acf (x) dx +  cb f (x) dx.
CHAPTER 2
2.4 Continuity
Example: Evaluate 2 [1 / (x (x2 - 4))] dx.
Example: Evaluate 0 x e-x dx.

9. CHAPTER 2
Comparison Theorem: Suppose that f and g are continuous functions with f(x)  g(x)  0 for x  a.
If af (x) dx is convergent, then ag(x) dx is convergent.
If ag(x) dx is divergent, then af (x) dx is divergent.
2.4 Continuity

10. CHAPTER 2
Example: Use the Comparison Theorem to determine whether 01[e-x/(x)] dx is divergent or convergent.
2.4 Continuity
Example: Sketch the region and find its area if the area is finite for S = {(x,y) | x  0, 0  y  1 / (x + 1) }.