8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 8.8 Improper Integrals 2015 Copyright © Cengage Learning. All rights reserved.

Warm-Up Evaluate the limit:

Objectives Evaluate an improper integral that has an infinite limit of integration. Evaluate an improper integral that has an infinite discontinuity.

Improper Integrals The definition of a definite integral requires that the interval [a, b] be finite, and that f be continuous on [a, b]. This section examines a procedure for evaluating integrals that do not satisfy these requirements—usually because either one or both of the limits of integration are infinite, or f has a finite number of infinite discontinuities in the interval [a, b]. Integrals that possess either property are said to be improper integrals.

Improper Integrals with Infinite Limits of Integration

Consider the integral This can be interpreted as the area of the shaded region in the figure: Now, taking the limit as The integral converges to 1.

Example 1 – An Improper Integral Evaluate Solution:

Example 1 – Solution cont’d

Example 2 – Improper Integrals Evaluate each improper integral: Solution:

Example 3 – Using L’Hopital’s with an Improper Integral Evaluate: Now apply the definition of an improper integral: Using L’Hopital’s on the limit:

Example 4 – Infinite upper and lower limits of integration Evaluate: Convenient breaking point? Solution:

You try Evaluate:

Improper Integrals with Infinite Limits of Integration

Homework Day 1: Section 8.8 pg. 585 #1-31 odd

Improper Integrals with Infinite Discontinuities Day 2 Improper Integrals with Infinite Discontinuities

Try this: Find the area of: Graph? To find the area under a curve that is discontinuous on the interval, divide the region at the discontinuity and compute the areas of the sub-regions. The area is infinite between -1 and 1.

Improper Integrals with Infinite Discontinuities A function f is said to have an infinite discontinuity at c if, from the right or left,

Improper Integrals with Infinite Discontinuities

Example 6 – An Improper Integral with an Infinite Discontinuity Evaluate Solution: The integrand has an infinite discontinuity at x = 0, as shown in Figure 8.24. You can evaluate this integral as shown below. Figure 8.24

Example 7 – An Improper Integral Evaluate Solution:

If you don’t check to see that the integral is improper, Example 8 – Try this one Evaluate Solution: The integral is improper because it has an infinite discontinuity at an interior point. Rewrite as If you don’t check to see that the integral is improper, you’ll arrive at 3/8, which is incorrect.

Example: (P is a constant.) What happens here? If then gets bigger and bigger as , therefore the integral diverges. If then b has a negative exponent and , therefore the integral converges.

Improper Integrals with Infinite Discontinuities

Example 11 – An Application Involving A Solid of Revolution The solid formed by revolving (about the x-axis) the unbounded region lying between the graph of f(x) = 1/x and the x-axis (x ≥ 1) is called Gabriel’s Horn. (See Figure 8.28.) Show that this solid has a finite volume and an infinite surface area. Figure 8.28

Example 11 – Solution Using the disk method and Theorem 8.5, you can determine the volume to be The surface area is given by

Example 11 – Solution Because cont’d Because on the interval [1, ), and the improper integral diverges, you can conclude that the improper integral also diverges. So, the surface area is infinite.

Homework Day 1: Section 8.8 pg. 585 #1-31 odd Day 2: Section 8.8 pg. 585 #33-49odd, 50, 53, 55, 67. Day 3: MMM 121,122,124