The exponential growth model is used to show how populations grow over time.  This model shows a population growing exponentially without a carrying capacity limiting the population at some point.   is the growth constant and  is the population.

The most simple exponential growth model only takes into account the population’s current state, , time, , and a growth/decay constant that is greater than zero .  It does not limit the population to a carrying capacity or take into account resource availability/ predator-prey interactions.

We will use separation of variables to solve this differential equation.

We use partial fractions for the left hand side:

It is clear that  so then  so our partial fraction decomposition:

Plugging back into our separation of variables:

                            Let 

                       Let 

We will evaluate at  in order to solve for .  Evaluating at  gives .  We will substitute this in for  into the equation we are solving.

Notice how the graph grows exponentially until it reaches a certain equilibrium.  This tells us that the population is approaching a carrying capacity so this graph shows logistic growth.  The other graph depicts exponential growth.

This example states a carrying capacity for the population.  Since exponential growth does not take into account carrying capacity, we cannot use this model for the population.  So we must use logistic growth.

The logistic growth model models a population taking into account a carrying capacity; that is, how large a population can grow and still survive off of available resources.  If the population is below this carrying capacity, it will grow to meet it.  If the population is higher than this carrying capacity, it will decrease to the carrying capacity.  This makes the carrying capacity a stable equilibrium point of the population.

This example only gives use a growth rate and a starting point for the population.  There is no limiting factor or carrying capacity so we must use exponential growth to model this population.

In this particular problem, the bounds for our integral are provided; specifically, we know that the integral setup should look something like . This can be determined by the given domain  and the fact that we are dealing with an area problem involving two functions. 

The next step is to continue creating the integral that will be used to evaluate the area. By graphing the functions, it can be determined that the function  takes on larger values of  than the other function , and therefore should be deemed the “upper function.”

Without graphing, this can also be determined because  will take on values between  and , while . 

The following integral is then created:

Finally, the definite integral is evaluated:

First, the intersection points of the two functions must be found in order to determine the bounds of the area integral we wish to set up. 

 and 

Next, the area integral must be set up correctly. To do this, the two functions must be either categorized as an “upper function” or a “lower function.” This is important as it determines which function comes first in our general formula for the area between two curves, . 

After examination either numerically or graphically,  is determined to be above the other function, . Now, the integral may be set up:

Finally, the integral must be evaluated:

This problem can be tricky to set up. There are two main methods for setting up definite integrals meant to evaluate the area between curves (see below). The setup is often dictated by the potential ease of evaluation.

OR 

Because one of the functions contains a  term, the expression may be easier to evaluate if we orient the integral in terms of . However, note that both methods are valid for this particular problem.

If the method chosen is the integral in terms of , the next step is determining the bounds. Remember that this process is the same as usual, but instead of bounds in terms of x we would like them in terms of .

Now, set the equations equal and solve for intersection points that will determine the bounds.

Finally, set up the integral with the format  and solve for the area.