Improper Integrals Objective: Evaluate integrals that become infinite within the interval of integration

Improper Integrals Our main objective in this section is to extend the concept of a definite integral to allow for infinite intervals of integration and integrands with vertical asymptotes within the interval of integration or one of the bounds. We will call the vertical asymptotes infinite discontinuities and we will call integrals with infinite intervals of integration or infinite discontinuities within the interval of integration improper integrals.

Improper Integrals Examples of such integrals are: Infinite intervals of integration.

Improper Integrals Examples of such integrals are: Infinite discontinuities in the interval of integration.

Improper Integrals Examples of such integrals are: Both Infinite discontinuities in the interval of integration and infinite intervals of integration.

Definition The improper integral of f over the interval is defined to be In the case where the limit exists, the improper integral is said to converge, and the limit is defined to be the value of the integral. In the case where the limit does not exist, the improper integral is said to diverge, and is not assigned a value.

Example 1a Evaluate

Example 1a Evaluate

Example 1a Evaluate

Example 1b Evaluate

Example 1b Evaluate

Example 1b Evaluate

Example 1c Evaluate

Example 1c Evaluate

Example 1c Evaluate

Improper Integrals These examples lead us to this theorem. if p > 1 if p < 1

Example 1 On the surface, the graphs of the last three examples seem very much alike and there is nothing to suggest why one of the areas should be infinite and the other two finite. One explanation is that 1/x 3 and 1/x 2 approach zero more rapidly than 1/x as x approaches infinity so that the area over the interval [1, b] accumulates less rapidly under the curves y = 1/x 3 and y = 1/x 2 than under y = 1/x. This slight difference is just enough that two areas are finite and one infinite.

Example 3 Evaluate

Example 3 Evaluate We need to use integration by parts.

Example 3 Evaluate We need to use integration by parts.

Example 3 Evaluate We need to use integration by parts.

Example 3 Evaluate

Example 3 Evaluate

Example 3 Evaluate This is of the form so we will use L’Hopital’s Rule

Example 3 Evaluate We can interpret this to mean that the net signed area between the graph of and the interval is 0.

Definition The improper integral of f over the interval is defined to be The integral is said to converge if the limit exists and diverge if it does not.

Definition The improper integral of f over the interval is defined to be where c is any real number (we will usually choose 0 to make it easier). The improper integral is said to converge if both terms converge and diverge if either term diverges.

Example 4 Evaluate

Example 4 Evaluate

Example 4 Evaluate

Example 4 Evaluate

Definition If f is continuous on the interval [a, b], except for an infinite discontinuity at b, then the improper integral of f over the interval [a, b] is defined as In the case where the limit exists, the improper integral is said to converge, and the limit is defined to be the value of the integral. If the limit does not exist, the integral is said to diverge.

Definition If f is continuous on the interval [a, b], except for an infinite discontinuity at a, then the improper integral of f over the interval [a, b] is defined as In the case where the limit exists, the improper integral is said to converge, and the limit is defined to be the value of the integral. If the limit does not exist, the integral is said to diverge.

Definition If f is continuous on the interval [a, b], except for an infinite discontinuity at a point c in (a, b), then the improper integral of f over the interval [a, b] is defined as The improper integral is said to converge if both terms converge and diverge if either term diverges.

Example 6a Evaluate

Example 6a Evaluate

Example 6a Evaluate

Example 6a Evaluate

Example 6b Evaluate

Example 6b Evaluate

Example 6b Evaluate

Example 6b Evaluate

Example 6b Evaluate

Example 6b Evaluate

Example 6b Evaluate

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