Study Guide: Improper Integrals

Learning Objectives

Evaluate definite integrals over infinite intervals using limits.
Identify and evaluate integrals of functions with infinite discontinuities on closed intervals.
Determine the convergence or divergence of an improper integral.
Apply the Comparison Theorem to determine the convergence behavior of complex improper integrals.

Key Terms & Glossary

Improper Integral: A definite integral that has either infinite limits of integration or an integrand with an infinite discontinuity.
Convergent Integral: An improper integral whose evaluation limit exists and yields a finite, real number.
Divergent Integral: An improper integral whose evaluation limit does not exist or approaches positive/negative infinity.
Infinite Discontinuity: A point where a function's value approaches infinity, resulting in a vertical asymptote on the graph.
Comparison Theorem: A method to determine the convergence of an improper integral by comparing it to a simpler, known integral.

The "Big Idea"

The fundamental "Big Idea" of Improper Integrals is the extension of the traditional definite integral to bounded limits. The Fundamental Theorem of Calculus requires a continuous function over a closed, finite interval $[a,b]$ . Improper integrals bypass this restriction by using limits to calculate the area under curves that stretch to infinity (horizontally) or spike to infinity (vertically). If the accumulated area settles on a finite value, the integral converges; if the area grows endlessly, it diverges.

Formula / Concept Box

[!IMPORTANT] Always replace the "problematic" bound (infinity or the point of discontinuity) with a variable, then take the limit as that variable approaches the bound.

Classification	Scenario	Limit Definition
Type 1	Upper bound is infinity	$\int_a^{\infty} f(x) dx = \lim_{t \to \infty} \int_a^t f(x) dx$
Type 1	Lower bound is infinity	$\int_{-\infty}^b f(x) dx = \lim_{t \to -\infty} \int_t^b f(x) dx$
Type 1	Both bounds infinite	$\int_{-\infty}^{\infty} f(x) dx = \int_{-\infty}^c f(x) dx + \int_c^{\infty} f(x) dx$
Type 2	Discontinuous at upper bound $b$	$\int_a^b f(x) dx = \lim_{t \to b^-} \int_a^t f(x) dx$
Type 2	Discontinuous at lower bound $a$	$\int_a^b f(x) dx = \lim_{t \to a^+} \int_t^b f(x) dx$
Type 2	Discontinuous at interior point $c$	$\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$

Hierarchical Outline

I. Integrating over an Infinite Interval (Type 1)
- A. Single Infinite Bound
  - Replace $\infty$ or $-\infty$ with $t.
  - Evaluate the definite integral from a $to$ t.
  - Take the limit as t \to \infty.
- B. Doubly Infinite Bounds
  - Split the integral at any convenient real number c $(often$ c = 0).
  - Rule: Both resulting integrals must converge for the whole integral to converge.
II. Integrating a Discontinuous Integrand (Type 2)
- A. Boundary Discontinuities
  - Identify vertical asymptotes at integration limits.
  - Approach the bound from the interior of the interval (e.g., t \to b^-).
- B. Interior Discontinuities
  - Split the integral directly at the discontinuity c.
  - Warning: Overlooking interior discontinuities is a highly common pitfall!
III. The Comparison Theorem
- A. Direct Comparison
  - Let f(x) $and$ g(x) $be continuous with $0 \le f(x) \le g(x)$ .
  - If larger area $\int g(x) dx$ converges, smaller area $\int f(x) dx MUST converge.
  - If smaller area \int f(x) dx $diverges, larger area$ \int g(x) dx$ MUST diverge.

Visual Anchors

Diagram 1: The Improper Integral Evaluation Process

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Diagram 2: Visualizing a Type 1 Integral

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Definition-Example Pairs

Term: Type 1 Improper Integral
- Definition: An integral where the domain of integration extends to infinity.
- Real-World Example: Calculating the total probability of an event happening over an indefinite future timeframe using a probability density function, such as finding the likelihood a lightbulb burns out eventually (from $t=0$ to $t=\infty).
Term: Type 2 Improper Integral
- Definition: An integral where the function possesses a vertical asymptote within the bounds of integration.
- Real-World Example: Calculating the gravitational potential energy of a particle moving extremely close to a point mass, where the force approaches infinity as the distance r$ approaches 0.
Term: Convergence
- Definition: The condition when the limit of the accumulated area under an improper curve results in a finite, specific value.
- Real-World Example: Reaching terminal velocity. As a skydiver falls (time approaches infinity), their speed converges to a maximum finite rate rather than increasing indefinitely.

Worked Examples

▶Example 1: Evaluating a Type 1 Integral (Infinite Limit)

Problem: Evaluate $\int_1^{\infty} \frac{1}{x^2} dx$ . State whether it converges or diverges.

Step 1: Rewrite as a limit. $\lim_{t \to \infty} \int_1^t x^{-2} dx$

Step 2: Find the antiderivative and evaluate bounds. $\lim_{t \to \infty} \left[ -\frac{1}{x} \right]_1^t$ $\lim_{t \to \infty} \left( -\frac{1}{t} - \left(-\frac{1}{1}\right) \right)$ $\lim_{t \to \infty} \left( 1 - \frac{1}{t} \right)$

Step 3: Evaluate the limit. As $t \to \infty$ , the term $\frac{1}{t}$ approaches 0. $1 - 0 = 1$

Conclusion: The integral converges to 1.

▶Example 2: Evaluating a Type 2 Integral (Discontinuous Integrand)

Problem: Evaluate $\int_0^1 \frac{1}{\sqrt{x}} dx$ . State whether it converges or diverges.

Step 1: Identify the discontinuity. The function $f(x) = \frac{1}{\sqrt{x}}$ is undefined (has a vertical asymptote) at $x = 0$ .

Step 2: Rewrite as a limit. $\lim_{t \to 0^+} \int_t^1 x^{-1/2} dx$

Step 3: Find the antiderivative and evaluate bounds. $\lim_{t \to 0^+} \left[ 2x^{1/2} \right]_t^1$ $\lim_{t \to 0^+} (2(1)^{1/2} - 2t^{1/2})$ $\lim_{t \to 0^+} (2 - 2\sqrt{t})$

Step 4: Evaluate the limit. As $t \to 0^+$ , the term $2\sqrt{t}$ approaches 0. $2 - 0 = 2$

Conclusion: The integral converges to 2.

[!TIP] Not all unbounded regions have infinite areas! Example 1 and 2 prove that geometric regions stretching to infinity can possess a finite, exact area.

Checkpoint Questions

What is the fundamental difference in setup between evaluating a Type 1 improper integral and a Type 2 improper integral?
If you are given the integral $\int_{-1}^1 \frac{1}{x^3} dx$ , what crucial step must you take before applying a limit?
According to the Comparison Theorem, if $f(x) \ge g(x) \ge 0$ and $\int g(x) dx diverges, what can be concluded about \int f(x) dx?
Why must we use limits to evaluate an integral that has a boundary at infinity, rather than simply plugging in "infinity" as a number?
If an improper integral is split into two parts (e.g., from -\infty $to 0 and 0 to$ \infty$), and the first part diverges while the second converges, what is the behavior of the overall integral?