Improper Integrals: Infinite Intervals and Discontinuous Integrands

Improper Integrals

Introduction to Improper Integrals

Improper integrals extend the concept of definite integrals to cases where the interval of integration is infinite or the integrand becomes unbounded within the interval. These integrals are essential for analyzing areas and quantities that would otherwise be undefined using standard integration techniques.

• Improper integrals of Type I involve infinite limits of integration.

• Improper integrals of Type II involve integrands with infinite discontinuities within the interval of integration.

Type I: Integrals Over Infinite Intervals

Improper integrals with infinite limits are defined using limits:

• If f(x) is continuous on [a, ∞), then: $\int_a^\infty f(x)\,dx = \lim_{t \to \infty} \int_a^t f(x)\,dx$

• If f(x) is continuous on (−∞, b], then: $\int_{-\infty}^b f(x)\,dx = \lim_{t \to -\infty} \int_t^b f(x)\,dx$

• If f(x) is continuous on (−∞, ∞), then: $\int_{-\infty}^{\infty} f(x)\,dx = \int_{-\infty}^c f(x)\,dx + \int_c^{\infty} f(x)\,dx$ where c is any real number.

If the limit exists and is finite, the improper integral converges; otherwise, it diverges.

Examples: Area Under Curves with Infinite Intervals

• Consider the region under $y = \frac{1}{x^2}$ for $x \geq 1$:

• The area from $x = 1$ to $x = t$ is: $A(t) = \int_1^t \frac{1}{x^2}\,dx = 1 - \frac{1}{t}$

• As $t \to \infty$, $A(t) \to 1$.

• For finite intervals, the area under $y = 1/x^2$ from $x = 1$ to $x = 2$, $3$, and $5$ are $1/2$, $2/3$, and $4/5$ respectively.

• As the upper limit increases, the area approaches 1.

Convergence and Divergence of Improper Integrals

• The integral $\int_1^\infty \frac{1}{x^2}\,dx$ converges to 1.

• The integral $\int_1^\infty \frac{1}{x}\,dx$ diverges (the area is infinite).

• For $\int_1^\infty \frac{1}{x^p}\,dx$:

  • Converges if $p > 1$

  • Diverges if $p \leq 1$

Type II: Integrals with Infinite Discontinuities

Improper integrals where the integrand becomes infinite at a point within the interval are defined as follows:

• If f(x) is continuous on (a, b] and discontinuous at a: $\int_a^b f(x)\,dx = \lim_{t \to a^+} \int_t^b f(x)\,dx$

• If f(x) is continuous on [a, b) and discontinuous at b: $\int_a^b f(x)\,dx = \lim_{t \to b^-} \int_a^t f(x)\,dx$

• If f(x) is discontinuous at c where $a < c < b$: $\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx$

If the limit exists and is finite, the improper integral converges; otherwise, it diverges.

Geometric Interpretation of Improper Integrals

• Improper integrals can represent the area under a curve over an infinite interval or where the curve has a vertical asymptote.

• For example, $\int_{-\infty}^{\infty} \frac{1}{1 + x^2}\,dx = \pi$ represents the total area under $y = \frac{1}{1 + x^2}$.

Comparison Tests for Improper Integrals

Direct Comparison Test

The Direct Comparison Test is used to determine the convergence or divergence of improper integrals by comparing them to known integrals.

• If $0 \leq f(x) \leq g(x)$ for all $x \geq a$:

• If $\int_a^\infty g(x)\,dx$ converges, then $\int_a^\infty f(x)\,dx$ also converges.

• If $\int_a^\infty f(x)\,dx$ diverges, then $\int_a^\infty g(x)\,dx$ also diverges.

Note: The converse is not necessarily true; if $\int_a^\infty f(x)\,dx$ converges, $\int_a^\infty g(x)\,dx$ may or may not converge.

Limit Comparison Test

If $f$ and $g$ are positive and continuous on $[a, \infty)$, and $\lim_{x \to \infty} \frac{f(x)}{g(x)} = L$ with $0 < L < \infty$, then both $\int_a^\infty f(x)\,dx$ and $\int_a^\infty g(x)\,dx$ either both converge or both diverge.

Summary Table: Convergence of $\int_1^\infty \frac{1}{x^p}\,dx$

Value of p	Convergence
$p > 1$	Converges
$p = 1$	Diverges
$p < 1$	Diverges

Key Points and Applications

• Improper integrals are evaluated using limits to handle infinite intervals or discontinuities.

• Convergence or divergence depends on the behavior of the integrand as $x$ approaches infinity or the point of discontinuity.

• Comparison tests are powerful tools for determining convergence when direct evaluation is difficult.

• Improper integrals have important applications in probability, physics, and engineering, such as calculating total probability, mass, or charge over unbounded domains.