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Calculus I · Unit 3A · lesson

Improper Integrals over Infinite Intervals

Concept

Learning objectives

Replace an infinite bound by a limit and determine convergence or divergence.

Improper Integrals over Infinite Intervals

Explanation

Infinity enters through a limit, not as an endpoint number

An integral such as ∫a∞f(x) dx\int_a^\infty f(x)\,dx is defined by replacing infinity with a finite boundary bb, evaluating ∫abf(x) dx\int_a^b f(x)\,dx, and then taking the limit as b→∞b\to\infty. The integral converges only if that limit exists and is finite. We never substitute infinity into an antiderivative as though it were an oversized real number.

Convergence depends on the long-run balance between interval length and function decay. A positive function may approach zero and still have an infinite total area if it decays too slowly. The classic pp-integrals show the threshold clearly. This distinction becomes important in probability, physics, and series theory, where an infinite domain can still carry a finite total mass or energy.

Define

∫a∞f(x) dx=lim⁡b→∞∫abf(x) dx,\int_a^\infty f(x)\,dx =\lim_{b\to\infty}\int_a^bf(x)\,dx,

provided the limit exists and is finite.

Worked example

A convergent tail

∫1∞1x2 dx=lim⁡b→∞[−1x]1b=lim⁡b→∞(1−1b)=1.\int_1^\infty\frac1{x^2}\,dx =\lim_{b\to\infty}\left[-\frac1x\right]_1^b =\lim_{b\to\infty}\left(1-\frac1b\right)=1.

Although the interval is infinite, the tail contributions shrink fast enough to produce a finite total.

Worked example

A divergent tail

∫1∞1x dx=lim⁡b→∞ln⁡b=∞.\int_1^\infty\frac1x\,dx =\lim_{b\to\infty}\ln b=\infty.

The integral diverges.

Interactive checku3a-improper-inf-01

Does ∫1∞x−3 dx\int_1^\infty x^{-3}\,dx converge or diverge?

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Show hint

Use the p-integral threshold: p>1p>1 converges.

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Tail area and convergence. Show finite truncations of an infinite-interval integral and a running total approaching a limit.
Read this graph as text

Tail area and convergence. The shaded tail extends right while accumulated area approaches 1. Show finite truncations of an infinite-interval integral and a running total approaching a limit. Do not portray infinity as a reachable endpoint.

Every relationship in tail area and convergence uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.

Why it matters: Show finite truncations of an infinite-interval integral and a running total approaching a limit.

Visual study

Tail area and convergence. Show finite truncations of an infinite-interval integral and a running total approaching a limit.

After the explanation

Use the section idea

Reading lens

Make approximation error and limiting behavior explicit rather than hiding them behind a calculator result or an infinity symbol.

Mental model

Numerical rules replace a curve with simple local shapes; improper integrals replace a forbidden endpoint or infinite interval with a limit.

Decision

Choose the rule and partition, estimate scale and sign, or write the correct defining limit before evaluating.

Common trap

Treating an approximation as exact, using Simpson's Rule with an invalid partition, or substituting infinity as though it were a number.

Check yourself

Can you defend the estimate's scale or the improper integral's convergence from a written calculation?

Source & rights

Original instruction with traceable references.

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