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

Abstract

This article presents rigorous step-by-step derivations of three foundational Calculus II techniques: logarithmic differentiation, convergence testing for improper integrals, and volume calculation for solids of revolution. Each derivation is motivated by the underlying mathematical structure and illustrated with worked examples. The goal is to provide a reference for students and instructors seeking transparent, technically sound explanations of these essential methods.

Background

Calculus II extends single-variable calculus into more sophisticated territory. Where Calculus I focuses on basic differentiation and integration, Calculus II demands fluency with advanced integration techniques, infinite series, and geometric applications. Three topics recur across most curricula: handling derivatives of complex functions via logarithmic methods, determining whether improper integrals converge, and computing volumes of three-dimensional solids formed by rotation.

This article treats each as a self-contained derivation, building from first principles and showing the algebraic steps that justify the final formulas.

Key Results

1. Logarithmic Differentiation

Problem: Differentiate a function $y = f(x)$ where $f$ is a product, quotient, or involves variable exponents—cases where the product or quotient rule becomes unwieldy.

Derivation ^{[logarithmic-differentiation]}

Begin with $y = f(x)$. Take 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: $\frac{d}{dx}[\ln(y)] = \frac{1}{y} \frac{dy}{dx}$

On the right: $\frac{d}{dx}[\ln(f(x))] = \frac{f'(x)}{f(x)}$

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

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

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

More usefully, for a composite function, this yields: $\frac{dy}{dx} = f(x) \cdot \frac{f'(x)}{f(x)}$

Why this works: Logarithms convert products into sums and exponents into coefficients, transforming a multiplicative problem into an additive one. This is especially powerful when the variable appears in both base and exponent.

2. Convergence of Improper Integrals

Problem: Determine whether an integral $\int_a^b f(x) \, dx$ converges (yields a finite value) or diverges (approaches infinity or fails to exist) when the limits are infinite or the integrand is unbounded.

Definition ^{[convergence-of-integrals]}

An integral $\int_a^b f(x) \, dx$ is said to converge if the limit of the integral exists and is finite as the bounds approach their limits. It diverges if the limit does not exist or is infinite.

For an improper integral with an infinite upper limit: $\int_a^{\infty} f(x) \, dx = \lim_{t \to \infty} \int_a^t f(x) \, dx$

The integral converges if and only if this limit is finite.

Comparison Test: If $0 \le f(x) \le g(x)$ for all $x \ge a$, then:

• If $\int_a^{\infty} g(x) \, dx$ converges, so does $\int_a^{\infty} f(x) \, dx$.
• If $\int_a^{\infty} f(x) \, dx$ diverges, so does $\int_a^{\infty} g(x) \, dx$.

Limit Comparison Test: If $f(x), g(x) > 0$ and $\lim_{x \to \infty} \frac{f(x)}{g(x)} = L$ where $0 < L < \infty$, then $\int_a^{\infty} f(x) \, dx$ and $\int_a^{\infty} g(x) \, dx$ either both converge or both diverge.

Intuition: These tests allow us to determine convergence without computing the integral explicitly, by comparing the given function to a known benchmark (often a power function or exponential).

3. Volume of a Solid of Revolution

Problem: Find the volume of a three-dimensional solid formed 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 $y = f(x)$ (above) and $y = g(x)$ (below), from $x = a$ to $x = b$. Rotate this region about the x-axis.

At each position $x$, the cross-section perpendicular to the x-axis is a disk (or washer) with:

• Outer radius: $R(x) = f(x)$
• Inner radius: $r(x) = g(x)$

The area of this washer is: $A(x) = \pi [R(x)^2 - r(x)^2] = \pi [f(x)^2 - g(x)^2]$

Integrate along the axis of rotation: $V = \int_a^b A(x) \, dx = \pi \int_a^b [f(x)^2 - g(x)^2] \, dx$

Shell Method (Rotation about the y-axis)

For rotation about the y-axis, consider cylindrical shells. At height $y$, a thin vertical strip of width $dy$ traces out a cylindrical shell with:

• Radius: $x(y)$ (distance from the y-axis)
• Height: $h(y) - k(y)$ (the width of the region at height $y$)
• Circumference: $2\pi x(y)$

The volume element is: $dV = 2\pi x(y) [h(y) - k(y)] \, dy$

Integrate: $V = 2\pi \int_c^d x(y) [h(y) - k(y)] \, dy$

Worked Examples

Example 1: Logarithmic Differentiation

Find $\frac{dy}{dx}$ for $y = x^x$.

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

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

Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = y[\ln(x) + 1] = x^x[\ln(x) + 1]$

Example 2: Convergence of an Improper Integral

Determine whether $\int_1^{\infty} \frac{1}{x^2 + 1} \, dx$ converges.

Solution: For large $x$, $\frac{1}{x^2 + 1} \approx \frac{1}{x^2}$.

We know $\int_1^{\infty} \frac{1}{x^2} \, dx$ converges (p-integral with $p = 2 > 1$).

Since $0 < \frac{1}{x^2 + 1} < \frac{1}{x^2}$ for all $x \ge 1$, by the Comparison Test, $\int_1^{\infty} \frac{1}{x^2 + 1} \, dx$ converges.

Example 3: Volume of a Solid of Revolution

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 with $f(x) = \sqrt{x}$ and $g(x) = 0$: $V = \pi \int_0^4 (\sqrt{x})^2 \, dx = \pi \int_0^4 x \, dx$

Evaluate: $V = \pi \left[ \frac{x^2}{2} \right]_0^4 = \pi \cdot \frac{16}{2} = 8\pi$

References

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

AI Disclosure

This article was drafted with the assistance of an AI language model. The mathematical content is derived from class notes and standard Calculus II pedagogy; all formulas and derivations have been verified against the source materials. The AI was used to organize the notes into a coherent narrative structure, paraphrase content, and format the mathematical exposition. A human author reviewed the final text for accuracy and clarity.