NCSSM Online AP Calculus (BC) Distance Education & Extended Programs    Lesson  1.5  ...  Improper  Integrals       INTRODUCTION In this lesson, we will extend our study of integration of continuous functions over finite intervals to that of functions that are not continuous over an interval and other integrands over infinite intervals. Consider a function that models the spread of a disease during an epidemic. Suppose the rate at which it spreads can be approximated by the function 𝑟 𝑡 = 1200𝑡𝑒!!.!!, where 𝑟 is measured in people/day, and 𝑡 is measured in days since the beginning of the epidemic. (Adapted from Single-Variable Calculus, by Hughes-Hallett, Gleason, McCallum, et al., 2002). Looking at the graph of this rate below, we notice that there is a horizontal asymptote of 𝑦 = 0. We can interpret this as saying that eventually, the spread of the disease slows down, and no new people will get sick. Given the rate of the spread of the disease, could we determine how many people get sick altogether? Yes, we can! We could integrate the rate function from 𝑡 = 0 through to infinity, to represent the entire duration of the spread of the disease. This would represent the area under the curve of the rate from 𝑡 = 0 through to infinity. This would give us the total number of people who eventually get sick from the beginning of the epidemic. This works…but how do we integrate “to infinity”? And how are we sure that the area under the curve isn’t infinite itself? This is what we’ll explore in this lesson.

NCSSM Online AP Calculus (BC) Distance Education & Extended Programs IMPROPER INTEGRALS WITH AN INFINITE LIMIT OF INTEGRATION Consider an integral with an infinite limit of integration 𝑓(𝑥)!

! 𝑑𝑥. While it may seem that these integrals would be infinite, it’s not always the case. Suppose that instead of looking at the entire area under the curve, we explored the area under the function 𝑓 over the interval [𝑎, 𝑏], where 𝑏 is some arbitrary value greater than 𝑎. Then, consider the value those areas approaches as 𝑏 gets larger and larger. If we can find a limit of those areas as 𝒃 approaches infinity, then we would actually have an area under the curve on the entire original interval. If that limit does not exist, then the integral would not be finite. This is the reasoning behind the following definitions:

   

Definitions:      

1. The  integral  of  a  function  𝑓  over  the  interval  [𝑎,∞)  is  defined  as  

𝑓(𝑥)!

!𝑑𝑥 = lim

!→!𝑓 𝑥 𝑑𝑥!

!.  

 2. The  integral  of  a  function  𝑓  over  the  interval  (−∞, 𝑏]  is  defined  as  

𝑓(𝑥)!

!!𝑑𝑥 = lim

!→!!𝑓 𝑥 𝑑𝑥!

!.  

 3. The  integral  of  a  function  𝑓  over  all  real  numbers  is  defined  as    

𝑓(𝑥)!

!!𝑑𝑥 = 𝑓 𝑥 𝑑𝑥

!

!!+ 𝑓 𝑥 𝑑𝑥

!

!,  

where  𝑐  is  some  real  number.    Notice  that  each  of  the  integrals  on  the  right  are  improper  and    would  be  evaluated  using  one  of  the  definitions  above.  

   

Any  one  of  these  integrals  diverges  if  a  limit  associated  in  the  evaluation  doesn’t  exist.      If  the  limit(s)  do  exist,  then  the  integral  converges  to  the  value  of  the  limit.  

     

Please watch the first example video we have provided for you in Canvas. Also see Examples 1-4 on Pages 549-551 of our Anton 10th Edition textbook.

NCSSM Online AP Calculus (BC) Distance Education & Extended Programs IMPROPER INTEGRALS WITH INTEGRANDS INCLUDING INFINITE DISCONTINUITIES Consider the function 𝑓 𝑥 = !

!!! on the interval (2, 4), as shown below:

We know that the function is not defined when 𝑥 = 2, and the graph contains a vertical asymptote there. If we were to consider the integral !

!!!!! 𝑑𝑥, this would represent the area under the curve 𝑓 𝑥 = !

!!!

between 𝑥 = 2 and 𝑥 = 4. This includes the discontinuity at 𝑥 = 2, and the region under the curve over this interval extends without bound in the vertical direction. As we did in our earlier discussion of improper integrals with an infinite limit of integration, we can instead consider the area under the curve over an interval [𝑎, 4], such that 𝑎 > 2, and determine whether that area approaches a finite value as 𝑎 approaches 2.

 Now consider this function over the interval [0, 4]. While the function is defined at each of the endpoints, it is not defined at 𝑥 = 2, which this interval includes. In a case such as this, if we’re interested in evaluating the integral of 𝑓 𝑥 = !

!!! from 𝑥 = 0 to 𝑥 = 4, we could split the integral into two in order to

account for the discontinuity at 𝑥 = 2. We could evaluate one from 𝑥 = 0 to 𝑥 = 2 and the other from 𝑥 = 2 to 𝑥 = 4. Each of these integrals would involve a limit of integration where the function is discontinuous; however, we can use the technique described above to evaluate each of them!

NCSSM Online AP Calculus (BC) Distance Education & Extended Programs

   

This is the reasoning behind the following definitions:

   

Definitions:      

1. The  integral  of  a  function  𝑓  over  the  interval  [𝑎, 𝑏],  where  the  function  has  one  infinite    discontinuity  at  𝑏,  is  defined  as    

𝑓(𝑥)!

!𝑑𝑥 = lim

!→!!𝑓 𝑥 𝑑𝑥!

!.  

 2. The  integral  of  a  function  𝑓  over  the  interval  [𝑎, 𝑏],  where  the  function  has  one  infinite  discontinuity  

at  𝑎,  is  defined  as    

𝑓(𝑥)!

!𝑑𝑥 = lim

!→!!𝑓 𝑥 𝑑𝑥!

!.  

 3. The  integral  of  a  function  𝑓  over  the  interval  [𝑎, 𝑏],  where  the  function  has  one  infinite  discontinuity  

within  the  interval  at  𝑐,  is  defined  as    

𝑓(𝑥)!

!𝑑𝑥 = 𝑓 𝑥 𝑑𝑥

!

!+ 𝑓 𝑥 𝑑𝑥

!

!,  

where  each  of  the  integrals  on  the  right  are  improper.    

 Any  one  of  these  integrals  diverges  if  a  limit  associated  in  the  evaluation  doesn’t  exist.    

 If  the  limit(s)  do  exist,  then  the  integral  converges  to  the  value  of  the  limit(s).     Please watch the second example video we have provided for you in Canvas. Also see Examples 5-6 on Pages 551-552 of our Anton 10th Edition textbook.