Calculus II: Worked Example Walkthroughs for Core Integration Techniques

Abstract

This article provides structured worked examples for three foundational Calculus II topics: convergence of improper integrals, volumes of solids of revolution, and logarithmic differentiation. Each section combines conceptual grounding with step-by-step solutions to illustrate how theoretical frameworks translate into practice. The goal is to bridge the gap between abstract definitions and computational fluency.

Background

Calculus II extends single-variable calculus into more sophisticated territory. While Calculus I focuses on limits, derivatives, and basic integration, Calculus II deepens integration techniques and introduces applications to geometry and analysis. Three topics recur across most curricula: determining whether improper integrals converge ^{[convergence-of-integrals]}, computing volumes when regions rotate about axes ^{[volume-of-solid-of-revolution]}, and differentiating functions with variable exponents via logarithmic methods ^{[logarithmic-differentiation]}.

Each topic requires both conceptual understanding and procedural skill. Worked examples serve as a bridge: they show not just what to do, but why each step follows logically from the previous one.

Key Results

Convergence of Improper Integrals

An integral converges when the limit of integration yields a finite value; it diverges otherwise ^{[convergence-of-integrals]}. This distinction matters because many real-world applications—probability distributions, work calculations, area under curves—depend on whether the integral is finite.

The most practical approach is to use comparison or limit comparison tests rather than compute the integral directly. These tests avoid tedious algebra while providing definitive answers.

Volumes of Solids of Revolution

When a region bounded by curves rotates about an axis, the resulting solid has volume given by integration ^{[volume-of-solid-of-revolution]}. The disk/washer method applies when rotating about the $x$- or $y$-axis; the shell method applies when the axis is parallel to the region's "natural" direction.

Logarithmic Differentiation

For functions where the variable appears in both base and exponent—or in complicated products—taking the natural logarithm before differentiating simplifies the algebra ^{[logarithmic-differentiation]}. The chain rule then yields the derivative without invoking the product rule repeatedly.

Worked Examples

Example 1: Testing Convergence of an Improper Integral

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

Solution:

This is an improper integral with an infinite upper limit. We cannot compute it directly, so we use the Limit Comparison Test ^{[convergence-of-integrals]}.

Step 1: Identify a comparison function.

For large $x$, the denominator $x^2 + x \approx x^2$, so: $\frac{1}{x^2 + x} \approx \frac{1}{x^2}$

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

Step 2: Apply the Limit Comparison Test.

Compute: $\lim_{x \to \infty} \frac{\frac{1}{x^2 + x}}{\frac{1}{x^2}} = \lim_{x \to \infty} \frac{x^2}{x^2 + x} = \lim_{x \to \infty} \frac{1}{1 + \frac{1}{x}} = 1$

Since the limit is a positive finite number and $\displaystyle \int_1^{\infty} \frac{1}{x^2} \, dx$ converges, the Limit Comparison Test guarantees that $\displaystyle \int_1^{\infty} \frac{1}{x^2 + x} \, dx$ also converges.

Conclusion: The integral converges.

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Example 2: Volume of a Solid of Revolution (Disk Method)

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:

We use the disk method ^{[volume-of-solid-of-revolution]}. When rotating about the $x$-axis, the volume is: $V = \pi \int_a^b [f(x)]^2 \, dx$

Step 1: Identify the bounds and the function.

The region extends from $x = 0$ to $x = 4$. The outer radius at position $x$ is $f(x) = \sqrt{x}$.

Step 2: Set up the integral.

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

Step 3: Evaluate.

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

Conclusion: The volume is $8\pi$ cubic units.

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Example 3: Logarithmic Differentiation

Problem: Find the derivative of $y = x^x$ for $x > 0$.

Solution:

The variable $x$ appears in both the base and exponent, so logarithmic differentiation is ideal ^{[logarithmic-differentiation]}.

Step 1: Take the natural logarithm of both sides.

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

Step 2: Differentiate both sides with respect to $x$.

Using the chain rule on the left and the product rule on the right: $\frac{1}{y} \frac{dy}{dx} = \ln(x) + x \cdot \frac{1}{x} = \ln(x) + 1$

Step 3: Solve for $\frac{dy}{dx}$.

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

Conclusion: $\displaystyle \frac{d}{dx}[x^x] = x^x(\ln(x) + 1)$.

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References

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

AI Disclosure

This article was drafted with AI assistance. The structure, worked examples, and explanations were generated from class notes using a language model. All mathematical claims are grounded in the cited notes; no results were invented. The author reviewed the content for accuracy and clarity before publication.