Calculus II: Worked Example Walkthroughs

Abstract

This article presents structured walkthroughs of three core Calculus II topics: logarithmic differentiation, volumes of solids of revolution, and convergence of improper integrals. Each section combines conceptual grounding with step-by-step examples designed to clarify technique and build intuition for students preparing for exams or independent study.

Background

Calculus II extends single-variable calculus into applications and advanced techniques. Three pillars emerge repeatedly: handling complex derivatives through algebraic transformation, computing volumes of three-dimensional objects via integration, and determining whether infinite integrals yield finite results. Mastery of these topics requires both procedural fluency and conceptual understanding ^{[logarithmic-differentiation]}, ^{[volume-of-solid-of-revolution]}, ^{[convergence-of-integrals]}.

This article assumes familiarity with basic differentiation, the chain rule, and antiderivatives. We work through representative examples to illustrate when and how to apply each technique.

Key Results

Logarithmic Differentiation

When a function involves a variable in both base and exponent, or consists of a complicated product or quotient, direct application of the product or quotient rule becomes unwieldy ^{[logarithmic-differentiation]}. Logarithmic differentiation transforms such problems by exploiting logarithm properties.

The core procedure: given $y = f(x)$, take the natural logarithm of both sides, differentiate implicitly, then solve for $\frac{dy}{dx}$ ^{[logarithmic-differentiation]}.

Why it works: Logarithms convert multiplication into addition and exponentiation into multiplication, simplifying the algebraic structure before differentiation.

Volumes of Solids of Revolution

Rotating a planar region around an axis generates a three-dimensional solid. The volume can be computed via integration ^{[volume-of-solid-of-revolution]}. Two standard setups:

• Rotation about the x-axis: $V = \pi \int_a^b (f(x)^2 - g(x)^2) \, dx$, where $f(x) \geq g(x)$ on $[a,b]$.
• Rotation about the y-axis: $V = 2\pi \int_c^d x (h(y) - k(y)) \, dy$, where $h(y)$ and $k(y)$ are the outer and inner radii as functions of $y$ ^{[volume-of-solid-of-revolution]}.

These formulas arise from the disk/washer method: slicing perpendicular to the axis of rotation yields circular cross-sections whose areas sum (via integration) to total volume.

Convergence of Improper Integrals

An improper integral has either infinite limits or an integrand with a singularity in the interval ^{[convergence-of-integrals]}. Such an integral converges if the limit of the integral as bounds approach their limits is finite; otherwise it diverges ^{[convergence-of-integrals]}.

Convergence tests—comparison, limit comparison, and integral tests—allow us to determine convergence without computing the integral explicitly ^{[convergence-of-integrals]}.

Worked Examples

Example 1: Logarithmic Differentiation

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

Solution:

Take the natural logarithm of both sides: $\ln(y) = \sin(x) \ln(x)$

Differentiate both sides with respect to $x$ using the product rule on the right: $\frac{1}{y} \frac{dy}{dx} = \cos(x) \ln(x) + \sin(x) \cdot \frac{1}{x}$

Multiply both sides by $y = x^{\sin x}$: $\frac{dy}{dx} = x^{\sin x} \left( \cos(x) \ln(x) + \frac{\sin(x)}{x} \right)$

Key insight: Direct differentiation using the chain rule would require careful tracking of exponent rules. Logarithmic differentiation ^{[logarithmic-differentiation]} converts the variable exponent into a product, making the chain rule straightforward.

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Example 2: 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:

The region is bounded above by $y = \sqrt{x}$ and below by $y = 0$, from $x = 0$ to $x = 4$. Rotating about the x-axis, we use the disk method ^{[volume-of-solid-of-revolution]}:

$V = \pi \int_0^4 (\sqrt{x})^2 - 0^2 \, dx = \pi \int_0^4 x \, dx$

Evaluate: $V = \pi \left[ \frac{x^2}{2} \right]_0^4 = \pi \left( \frac{16}{2} - 0 \right) = 8\pi$

Key insight: The formula $V = \pi \int_a^b (f(x)^2 - g(x)^2) \, dx$ applies directly when rotating about the x-axis. The squared terms represent the areas of circular cross-sections at each $x$.

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Example 3: Convergence of an Improper Integral

Problem: Determine whether $\int_1^{\infty} \frac{1}{x^2} \, dx$ converges or diverges.

Solution:

This is an improper integral with an infinite upper limit. We evaluate it as a limit ^{[convergence-of-integrals]}:

$\int_1^{\infty} \frac{1}{x^2} \, dx = \lim_{t \to \infty} \int_1^t x^{-2} \, dx$

Compute the antiderivative: $\int x^{-2} \, dx = -x^{-1} = -\frac{1}{x}$

Evaluate from $1$ to $t$: $\lim_{t \to \infty} \left[ -\frac{1}{x} \right]_1^t = \lim_{t \to \infty} \left( -\frac{1}{t} + 1 \right) = 0 + 1 = 1$

Since the limit is finite, the integral converges to $1$ ^{[convergence-of-integrals]}.

Key insight: For integrals of the form $\int_1^{\infty} \frac{1}{x^p} \, dx$, convergence depends on whether $p > 1$ (converges) or $p \leq 1$ (diverges). This is a standard benchmark for comparison tests ^{[convergence-of-integrals]}.

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References

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

AI Disclosure

This article was drafted with the assistance of an AI language model. The mathematical content and examples are derived from class notes and standard Calculus II pedagogy. All claims are cited to source notes. The article has been reviewed for accuracy and clarity by the author.