May 26, 2026 - Study Notes on Improper Integrals and Infinite Boundaries (Concise)

Conceptual Definition: An improper integral occurs when the interval of integration is infinite or when the integrand has an infinite discontinuity on the interval.
Notation and Limits: To evaluate an integral with an infinite bound, replace the infinity symbol with a variable $t$ . For example, $\int_a^{\infty} f(x) \,dx = \lim_{t \rightarrow \infty} \int_a^t f(x) \,dx$ .
Continuity Requirement: For these methods to apply, $f(x)$ must be continuous on the interval $[a, \infty)$ . If there are breaks or jumps, it is a different type of improper integral.
Convergence vs. Divergence:
- Convergent: If the limit exists and results in a finite value.
- Divergent: If the limit results in infinity, negative infinity, or does not exist.

Comparative Check ( $1/x$ vs. $1/x^2$ ):
- $\int_1^{\infty} \frac{1}{x} \,dx = \lim_{t \rightarrow \infty} [\ln(x)]_1^t = \lim_{t \rightarrow \infty} \ln(t) = \infty$ . This integral diverges.
- $\int_1^{\infty} \frac{1}{x^2} \,dx$ results in a finite value ( $1$ ). This integral converges.
General Rule: For integrals of the form $\int_a^{\infty} \frac{1}{x^p} \,dx$ , the integral will converge if $p > 1$ , as the anti-derivative will retain an $x$ in the denominator, which goes to zero as $x \rightarrow \infty$ .
Gabriel’s Horn: A specialized example involving revolving $f(x) = \frac{1}{x}$ around the $x$ -axis.
- Volume: $V = \int_1^{\infty} \pi \left(\frac{1}{x}\right)^2 \,dx = \pi$ . The volume is finite.
- Surface Area: The surface area is infinite, creating a mathematical paradox where one could fill the object with paint but never finish painting the outside.

Splitting Integrals: If an integral ranges from $-\infty$ to $\infty$ , it must be split into two separate limits at a point within the domain (usually $0$ ): $\int_{-\infty}^{\infty} f(x) \,dx = \int_{-\infty}^{0} f(x) \,dx + \int_{0}^{\infty} f(x) \,dx$
Symmetry Properties:
- Even Functions: If $f(x)$ is even, the integral over $(-\infty, \infty)$ is twice the integral over $[0, \infty)$ .
- Odd Functions: If $f(x)$ is odd (e.g., $x^3$ ), the area on the left and right sides of the $y$ -axis cancel out, resulting in a total area of zero.

Boundary Discontinuities: If $f(x)$ is discontinuous at a bound (e.g., at $x=b$ ), use a one-sided limit: $\lim_{t \rightarrow b^-} \int_a^t f(x) \,dx$ .
Interior Discontinuities: if $f(x)$ is discontinuous at a point $c$ between $a$ and $b$ , split the integral at $c$ and take the limit from both the left and the right.
Example ( $\ln(x)$ ): Finding the area under $x^2 \ln(x)$ from $0$ to $2$ requires a limit as $t \rightarrow 0^+$ because $\ln(x)$ is undefined at zero.

L’Hôpital’s Rule: Often required when evaluating limits resulting in indeterminate forms like $\frac{\infty}{\infty}$ or $0 \times \infty$ . For products, rewrite the function as a fraction before applying the derivative rule.
Trigonometric Limits:
- $\int \frac{1}{x^2 + a^2} \,dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right)$ .
- Key values: $\lim_{x \rightarrow \infty} \arctan(x) = \frac{\pi}{2}$ and $\lim_{x \rightarrow -\infty} \arctan(x) = -\frac{\pi}{2}$ .

Question: Does an improper integral always converge?
Answer: No. It depends on the specific curve and whether the anti-derivative grows or stabilizes as it approaches the limit.
Question: In the Gabriel's Horn example, are we using the washer method?
Answer: No, this is the disc method. We will discuss the disc method formally after the first test.
Question: Is it because tangent is negative that it flips the graph?
Answer: No, that is a property of inverses. Inverse functions reflect around the line $y = x$ , which flips both the orientation and the axes simultaneously.