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

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

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.

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

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?

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

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.

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.

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

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) }.