Previously, we defined the definite integral with the assumption that the function f is continuous on a closed interval.

Now, we shall look at definite integrals that have infinite limits of integration.

Such integrals are known as improper integrals.

Such integrals are known as improper integrals.

Use the slider to vary the value of the upper limit of integration b. Notice how the calculated area under the curve changes.

We can calculate the area for a closed interval by evaluating the given definite integral.

Now, suppose we have an infinite limit of integration.

Which of the following do you think is the value of this integral? Click the "Submit" button after selecting your answer.

We cannot evaluate this improper integral by the direct application of the fundamental theorem.

However, we can express this integral as the limit of an integral that can be evaluated.

After evaluating the integral, we take the limit of the result.

This gives us the solution to the original improper integral.

We now have the means to evaluate improper integrals with infinite limits. We can define three cases of improper integrals. The first has only an infinite upper limit of integration.

The second case involves an integral that has only an infinite lower limit of integration.

The third case involves an integral where both limits of integration are infinite.

Consider this function and its graph.

The area under the curve from x equals one to infinity can be calculated from the given improper integral.

What is the area given by this integral? Click the "Submit" button after entering your answer.

The limit of the integral evaluates to a real value.

Now, consider this improper integral. It is similar in form to our previous example.

In such a case where the limit exists, we say that the improper integral converges.

Let's apply the definition of the improper integral.

What is the solution of the integral? Click the "Submit" button after selecting your answer.

After solving the integral, we see that the limit of the result does not exist.

In this case, we say the improper integral diverges.

Let's look at the graphs of the integrands in our last two examples.

We see that one approaches the x-axis much more rapidly. This is also the one whose integral converges.

The two integrands have a similar form. In general, an improper integral having this form converges if the power of x is greater than one.

It diverges if the power of x is less than or equal to one.

We can illustrate this with a table showing different integrals of the same form that are evaluated with increasing upper limits of integration.

Now, consider this function and its graph.

We wish to determine the area under the curve given by this improper integral. Notice it has an infinite lower limit of integration.

What is the area? Click the "Submit" button after selecting your answer.

The limit of the integral exists. Therefore, this improper integral converges.

Previously, we determined the area under the curve in this graph from zero to infinity.

Now, let's consider the area under the entire curve.

We can determine this area using the given improper integral. In this case, both limits of integration are infinite.

What is value of this integral? Click the "Submit" button after selecting your answer.

The area we calculate under the entire curve is twice that we calculated for the interval zero to infinity.

This makes sense considering the curve is symmetric about the y-axis.

What can be said about the value of the area as the value of b is increased? Click the "Submit" button after selecting your answer.

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