Calculus II: Step-by-Step Derivations of Core Techniques

Abstract

This article presents rigorous derivations of three foundational Calculus II techniques: logarithmic differentiation, convergence analysis of improper integrals, and volume computation for solids of revolution. Each section develops the mathematical framework from first principles, with worked examples demonstrating practical application. The goal is to provide a reference for students and practitioners seeking clarity on the logical structure underlying these methods.

Background

Calculus II extends single-variable calculus by introducing techniques for handling complex functions, infinite domains, and geometric applications. Three problems recur throughout the course:

1. Differentiation of complicated functions — particularly those with variable exponents or nested operations
2. Evaluation of improper integrals — determining whether integrals over infinite intervals or with singularities yield finite values
3. Geometric computation — finding volumes of three-dimensional objects defined by rotation

This article addresses each systematically.

Key Results

Logarithmic Differentiation

Problem: Differentiate functions where the variable appears in both base and exponent, or where products and quotients make direct application of the product rule unwieldy.

Derivation: ^{[logarithmic-differentiation]}

Suppose $y = f(x)$ is a differentiable function. Taking the natural logarithm of both sides:

$\ln(y) = \ln(f(x))$

Differentiate both sides with respect to $x$ using the chain rule on the left and the chain rule on the right:

$\frac{1}{y} \frac{dy}{dx} = \frac{f'(x)}{f(x)}$

Multiply both sides by $y$:

$\frac{dy}{dx} = y \cdot \frac{f'(x)}{f(x)}$

Substitute $y = f(x)$ back:

$\frac{dy}{dx} = f(x) \cdot \frac{f'(x)}{f(x)}$

Key insight: Taking logarithms converts multiplicative structure into additive structure, allowing the chain rule to handle exponents and products more cleanly than direct application of the product or quotient rule.

Convergence of Improper Integrals

Problem: Determine whether an integral $\int_a^b f(x) \, dx$ yields a finite value when the domain is infinite or the integrand has singularities.

Definition: ^{[convergence-of-integrals]}

An integral is convergent if the limit of the integral as the bounds approach their limits exists and is finite. Formally, for an integral with an infinite upper bound:

$\int_a^{\infty} f(x) \, dx = \lim_{t \to \infty} \int_a^t f(x) \, dx$

If this limit exists and is finite, the integral converges. If the limit is infinite or does not exist, the integral diverges.

Intuition: Convergence determines whether the "total area" under a curve is finite. This is essential in applications ranging from probability (where integrals must equal 1) to physics (where infinite energy or work signals unphysical behavior).

Comparison Test: To avoid computing the integral directly, compare $f(x)$ to a known function $g(x)$:

• If $0 \le f(x) \le g(x)$ and $\int_a^{\infty} g(x) \, dx$ converges, then $\int_a^{\infty} f(x) \, dx$ converges.
• If $f(x) \ge g(x) \ge 0$ and $\int_a^{\infty} g(x) \, dx$ diverges, then $\int_a^{\infty} f(x) \, dx$ diverges.

Volume of Solids of Revolution

Problem: Find the volume of a three-dimensional object obtained by rotating a two-dimensional region around an axis.

Disk Method (Rotation about the x-axis): ^{[volume-of-solid-of-revolution]}

Consider a region bounded by $y = f(x)$ (upper curve) and $y = g(x)$ (lower curve) from $x = a$ to $x = b$. Rotating this region about the x-axis generates a solid. Slice the solid perpendicular to the x-axis at position $x$. The cross-section is a washer (annulus) with:

• Outer radius: $R(x) = f(x)$
• Inner radius: $r(x) = g(x)$
• Area of washer: $A(x) = \pi R(x)^2 - \pi r(x)^2 = \pi(f(x)^2 - g(x)^2)$

Integrate the cross-sectional areas:

$V = \int_a^b \pi(f(x)^2 - g(x)^2) \, dx = \pi \int_a^b (f(x)^2 - g(x)^2) \, dx$

Shell Method (Rotation about the y-axis): For rotation about the y-axis, use cylindrical shells. A thin vertical strip at position $x$ with thickness $dx$ and height $h(x)$ traces a cylindrical shell with:

• Radius: $x$
• Height: $h(x)$
• Circumference: $2\pi x$
• Surface area: $2\pi x \cdot h(x)$
• Volume element: $dV = 2\pi x \cdot h(x) \, dx$

Integrate:

$V = 2\pi \int_a^b x \cdot h(x) \, dx$

Worked Examples

Example 1: Logarithmic Differentiation

Problem: Differentiate $y = x^x$.

Solution: Take the natural logarithm: $\ln(y) = \ln(x^x) = x \ln(x)$

Differentiate both sides: $\frac{1}{y} \frac{dy}{dx} = \ln(x) + x \cdot \frac{1}{x} = \ln(x) + 1$

Multiply by $y$: $\frac{dy}{dx} = x^x(\ln(x) + 1)$

Example 2: Convergence of an Improper Integral

Problem: Does $\int_1^{\infty} \frac{1}{x^2} \, dx$ converge?

Solution: Compute the limit: $\int_1^{\infty} \frac{1}{x^2} \, dx = \lim_{t \to \infty} \int_1^t x^{-2} \, dx = \lim_{t \to \infty} \left[-\frac{1}{x}\right]_1^t = \lim_{t \to \infty} \left(-\frac{1}{t} + 1\right) = 1$

The integral converges to 1.

Example 3: Volume of a Solid of Revolution

Problem: Find the volume of the solid obtained by rotating the region bounded by $y = \sqrt{x}$, $y = 0$, and $x = 4$ about the x-axis.

Solution: Using the disk method: $V = \pi \int_0^4 (\sqrt{x})^2 \, dx = \pi \int_0^4 x \, dx = \pi \left[\frac{x^2}{2}\right]_0^4 = \pi \cdot 8 = 8\pi$

References

• ^{[logarithmic-differentiation]}
• ^{[convergence-of-integrals]}
• ^{[volume-of-solid-of-revolution]}

AI Disclosure

This article was drafted with AI assistance. The mathematical derivations and worked examples were synthesized from class notes and standard Calculus II pedagogy. All claims are grounded in the cited notes. The author reviewed the mathematical content for accuracy and clarity.