In this section, we will define what it means for an integral to be improper and begin investigating how to determine convergence or divergence of such integrals. Section 10.1 Improper Integrals
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In this section, we will define what it means for an integral to be improper and begin investigating how to determine convergence or divergence of such integrals.
Section 10.1 Improper Integrals
Idea
All of our definite integrals thus far have dealt with a function f that is continuous on the closed, bounded interval [a, b].
If an infinite discontinuity existed in [a, b], or the interval itself was unbounded, then we have an improper integral.
Infinite Intervals
We might have one (or both) of the limits of integration being infinite.
Infinite Integrands
The function could have an infinite discontinuity somewhere in the interval [a, b].
Approach
We will use limits.
If exists, then we say I
converges to this value, otherwise it diverges.
Approach
Suppose f has an infinite discontinuity at x = a.
If exists, then we say I
converges to this value, otherwise it diverges.
Suppose f has an infinite discontinuity at x = b.
If exists, then we say I
converges to this value, otherwise it diverges.
Example 1
Determine whether the integral converges or diverges. If it converges, state to what value.
Example 2
Determine whether the integral converges or diverges. If it converges, state to what value.
Example 3
Determine whether the integral converges or diverges. If it converges, state to what value.
Example 4
Determine whether the integral converges or diverges. If it converges, state to what value.
Example 5
Determine whether the integral converges or diverges. If it converges, state to what value.
Example 6
Determine whether the integral converges or diverges. If it converges, state to what value.
Example 7
Determine whether the integral converges or diverges. If it converges, state to what value.
Example 8
Determine whether the integral converges or diverges. If it converges, state to what value.
Example 9
Determine whether the integral converges or diverges. If it converges, state to what value.
Example 10
Determine whether the integral converges or diverges. If it converges, state to what value.
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