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CalculusBy Educator Today

C28 · Unit C6 Integration and accumulation

Improper integrals and convergence

BC extensionReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 6.13–6.14 · BC
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

An infinite bound or unbounded integrand makes an integral improper. Replace each problematic endpoint with a variable cutoff and take a one-sided limit. If a singularity lies inside an interval, split there and require both improper pieces to converge separately. Cancellation of opposite infinities is not convergence. For ∫ sub 1 ∞x^( minus p)dx, convergence occurs exactly when p>1.

Worked example

∫ sub 1 ∞1 over x squared dx equals lim as b tends to ∞[-1 over x] sub 1 ᵇ equals lim(1 minus 1 over b) equals 1. But ∫ sub 1 ∞1 over x dx equals lim ln b diverges. Having an integrand that tends to zero is necessary in many common examples but not a sufficient convergence test.

Explore the mathematics

Improper integrals and convergence

Replace an infinite endpoint with a limiting process.

Setting 3 of 5

Scroll sideways to see the full diagram.

An infinite endpoint requires a limit; do not substitute infinity as a number.xy1∫₁^∞ x^(−4) dx

Predict before continuing

An infinite endpoint requires a limit; do not substitute infinity as a number.

Read the complete explanation
  1. An infinite endpoint requires a limit; do not substitute infinity as a number.
  2. First integrate on [1,b], with finite b.
  3. For p>1 the upper-bound term tends to zero as b→∞.
  4. With p=4, the integral is 1/(p−1)=0.3333. For p≤1 this p-integral diverges.
See the idea: Improper integrals and convergence
See the idea: Improper integrals and convergenceThe lower curve 1/x² has finite total area from 1 to infinity, equal to 1. The upper curve 1/x also tends to zero but its improper integral diverges. A finite plot alone cannot decide tail convergence.xy102.750.34.50.66.250.981.2

The lower curve 1/x² has finite total area from 1 to infinity, equal to 1. The upper curve 1/x also tends to zero but its improper integral diverges. A finite plot alone cannot decide tail convergence.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Infinity is not an endpoint to substitute as an ordinary number. A finite principal value is not the same as convergence of an improper integral.

Investigate and explain

Compare x^( minus p) near zero and near infinity. Derive why the convergence inequalities reverse.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.