Calculus II: Worked Example Walkthroughs for Convergence, Differentiation, and Solids of Revolution

Abstract

This article provides structured worked examples for three core Calculus II topics: convergence of improper integrals, logarithmic differentiation, and volumes of solids of revolution. Each section combines conceptual grounding with step-by-step problem solutions, designed to reinforce both procedural fluency and conceptual understanding. The examples are drawn from typical exam-level problems and illustrate when and how to apply each technique.

Background

Calculus II extends single-variable calculus into applications and advanced techniques. Three topics form a conceptual backbone: determining whether integrals converge to finite values, differentiating complex functions via logarithmic methods, and computing volumes of three-dimensional objects formed by rotation. Each requires careful setup, correct application of rules, and verification of results.

Convergence of Integrals

An integral is said to converge when the limit of integration yields a finite value; it diverges when the limit is infinite or does not exist ^{[convergence-of-integrals]}. This distinction matters because improper integrals—those with infinite bounds or discontinuous integrands—require explicit limit evaluation. Without convergence tests, one cannot safely claim a result is meaningful ^{[convergence-of-integrals]}.

Logarithmic Differentiation

When a function involves a variable in both base and exponent, or is a complicated product or quotient, logarithmic differentiation offers a cleaner path than the product or quotient rule ^{[logarithmic-differentiation]}. The method transforms multiplicative structure into additive structure via the logarithm, simplifying chain rule application ^{[logarithmic-differentiation]}.

Solids of Revolution

Rotating a planar region about an axis generates a three-dimensional solid. Integration allows us to sum infinitesimal disk or washer cross-sections to find total volume ^{[volume-of-solid-of-revolution]}. This technique underpins engineering and physics applications where rotationally symmetric shapes are common ^{[volume-of-solid-of-revolution]}.

Key Results

Convergence Test Framework

For an improper integral $\int_a^{\infty} f(x) \, dx$, evaluate: $\lim_{t \to \infty} \int_a^t f(x) \, dx$

If the limit is finite, the integral converges; otherwise it diverges ^{[convergence-of-integrals]}.

Logarithmic Differentiation Formula

For $y = f(x)$, take the natural logarithm of both sides: $\ln(y) = \ln(f(x))$

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

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

^{[logarithmic-differentiation]}

Volume of Solid of Revolution (Disk/Washer Method)

Rotating the region between $y = f(x)$ and $y = g(x)$ (where $f(x) \geq g(x)$) about the x-axis from $x = a$ to $x = b$: $V = \pi \int_a^b \left[f(x)^2 - g(x)^2\right] \, dx$

For rotation about the y-axis: $V = 2\pi \int_c^d x \left[h(y) - k(y)\right] \, dy$

where $h(y)$ and $k(y)$ are the outer and inner radii as functions of $y$ ^{[volume-of-solid-of-revolution]}.

Worked Examples

Example 1: Convergence of an Improper Integral

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

Solution:

Set up the limit definition ^{[convergence-of-integrals]}: $\int_1^{\infty} \frac{1}{x^2} \, dx = \lim_{t \to \infty} \int_1^t x^{-2} \, dx$

Evaluate the antiderivative: $\int_1^t x^{-2} \, dx = \left[-x^{-1}\right]_1^t = -\frac{1}{t} - \left(-\frac{1}{1}\right) = 1 - \frac{1}{t}$

Take the limit: $\lim_{t \to \infty} \left(1 - \frac{1}{t}\right) = 1$

Conclusion: The integral converges to $1$ ^{[convergence-of-integrals]}.

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Example 2: Logarithmic Differentiation

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

Solution:

Take the natural logarithm of both sides ^{[logarithmic-differentiation]}: $\ln(y) = \ln(x^x) = x \ln(x)$

Differentiate both sides with respect to $x$ using the chain rule on the left and the product rule on the right: $\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}[x \ln(x)] = 1 \cdot \ln(x) + x \cdot \frac{1}{x} = \ln(x) + 1$

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

Verification: For $x = 1$, we get $\frac{dy}{dx} = 1^1[\ln(1) + 1] = 1 \cdot 1 = 1$, which is consistent with the behavior of $y = x^x$ near $x = 1$.

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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:

Identify the bounds and functions. The region is bounded above by $y = \sqrt{x}$ and below by $y = 0$ (the x-axis), from $x = 0$ to $x = 4$.

Apply the disk method formula for rotation about the x-axis ^{[volume-of-solid-of-revolution]}: $V = \pi \int_0^4 \left[(\sqrt{x})^2 - 0^2\right] \, dx = \pi \int_0^4 x \, dx$

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

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

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Example 4: Washer Method

Problem: Find the volume of the solid obtained by rotating the region bounded by $y = x$ and $y = x^2$ about the x-axis.

Solution:

First, find the intersection points: $x = x^2 \Rightarrow x(1 - x) = 0 \Rightarrow x = 0$ or $x = 1$. For $0 \leq x \leq 1$, we have $x \geq x^2$.

Apply the washer method ^{[volume-of-solid-of-revolution]}: $V = \pi \int_0^1 \left[x^2 - (x^2)^2\right] \, dx = \pi \int_0^1 \left[x^2 - x^4\right] \, dx$

Evaluate: $V = \pi \left[\frac{x^3}{3} - \frac{x^5}{5}\right]_0^1 = \pi \left(\frac{1}{3} - \frac{1}{5}\right) = \pi \cdot \frac{5 - 3}{15} = \frac{2\pi}{15}$

Conclusion: The volume is $\frac{2\pi}{15}$ cubic units.

References

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

AI Disclosure

This article was drafted with the assistance of an AI language model. The mathematical content, structure, and worked examples were generated based on class notes and standard Calculus II pedagogy. All claims are cited to source notes. The article has been reviewed for mathematical accuracy and clarity but should be verified against your course materials and textbooks before use in academic contexts.