8.4 Improper Integrals

ln 2 0 (-2,2)

Until now we have been finding integrals of continuous functions over closed intervals. Sometimes we can find integrals for functions where the function or the limits are infinite. These are called improper integrals.

Example 1: The function is undefined at x = 1. Since x = 1 is an asymptote, the function has no maximum. Can we find the area under an infinitely high curve? We could define this integral as: (left hand limit) We must approach the limit from inside the interval.

Rationalize the numerator.

This integral converges because it approaches a solution.

Example 2: This integral diverges. (right hand limit) We approach the limit from inside the interval.

Example 3: The function approaches when.

Example 4: 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. (P is a constant.) 

8.4 Tests for Convergence

Converges

Does converge? Compare: to for positive values of x. For

Since is always below, we say that it is “bounded above” by. Since converges to a finite number, must also converge!

Direct Comparison Test: Let f and g be continuous on with for all, then: 2 diverges if diverges. 1 converges if converges.

Example 7: The maximum value of so: on Since converges, converges.

Example 7: for positive values of x, so: Since diverges, diverges. on

If functions grow at the same rate, then either they both converge or both diverge. Does converge? As the “1” in the denominator becomes insignificant, so we compare to. Since converges, converges.

Of course



Gabriel’s Horn: Find the volume and the surface area.

Gabriel’s Horn: Find the volume and the surface area. It’s a solid with infinite surface area wrapped around a finite volume!!

Evaluate the improper integral or state that it diverges.

Evaluate the improper integral:

Evaluating an integral on Evaluate the improper integral.

Evaluating an integral on

Interior Infinite Discontinuity State why the integral is improper. Then evaluate the integral or state that it diverges.

Interior Infinite Discontinuity