Consider this function and its graph.

The function is discontinuous at x equals zero.

Now, suppose we wish to determine the area under the curve from zero to one.

The area cannot be determined directly from the given integral because the integrand is discontinuous.

In order to evaluate this integral, we express it as the limit of an integral that can be evaluated.

The integral in this example represents another type of improper integral...one that has a discontinuous integrand.

We can define three cases of improper integrals with discontinuous integrands. In the first case, the integrand is discontinuous at the lower limit of integration.

This is illustrated by the given graph of the integrand function. The curve is discontinuous at x equals a.

The second case involves an improper integral with an integrand that is discontinuous at the upper limit of integration.

We see in the graph that the integrand function curve is discontinuous at x equals b.

In the third case, the integrand is discontinuous at an interior point of the interval of integration.

As before, an improper integral of this type converges if the definition limits exist.

If the definition limits do not exist, the improper integral diverges.

Now, consider this definite integral.

At what point is the integrand discontinuous? Click the "Submit" button after selecting your answer.

As we see in this graph of the integrand function, the integrand is discontinuous at the upper limit of integration.

Therefore, we can evaluate the integral using the second case of our definition of improper integrals with discontinuous integrands.

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

Looking at this definite integral, it seems it can be evaluated quite easily in a direct manner.

However, if we graph the integrand function, we see that the curve is discontinuous at a point within the given interval of integration.

It is important that we recognize improper integrals, so that they can be evaluated properly. Drawing a graph of the integrand function can be very helpful in this respect.

We must try to evaluate this improper integral using the third case of the definition.

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

Not all of the limits can be evaluated. Therefore, the integral diverges.

Now, what is the area for the given interval? Click the "Submit" button after selecting your answer.

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