Calculus II: Worked Example Walkthroughs for Integration and Differentiation

Abstract

This article presents structured walkthroughs of three core Calculus II topics: convergence of improper integrals, logarithmic differentiation, and volumes of solids of revolution. Each section combines conceptual grounding with concrete worked examples to reinforce problem-solving technique and build intuition for when and how to apply these methods.

Background

Calculus II extends single-variable calculus into techniques for handling more complex functions and infinite domains. Three recurring challenges in this course are: determining whether integrals with infinite bounds or discontinuous integrands yield finite values; differentiating functions where the variable appears in both base and exponent; and computing volumes of three-dimensional objects formed by rotating planar regions. This article addresses each through explanation and example.

Key Results

Convergence of Improper Integrals

An integral is said to be convergent if the limit of the integral as the bounds approach their limits of integration exists and is finite ^{[convergence-of-integrals]}. Conversely, an integral diverges if this limit does not exist or is infinite. This distinction matters because many real-world applications—calculating areas under curves, computing probabilities, or modeling physical quantities—require knowing whether the result is meaningful (finite) or unbounded.

When the integrand approaches infinity at a point within the interval or the limits of integration are infinite, standard integration rules do not directly apply. In these cases, convergence tests such as the comparison test or limit comparison test provide systematic ways to determine behavior without computing the integral explicitly ^{[convergence-of-integrals]}.

Logarithmic Differentiation

Logarithmic differentiation is a technique for differentiating functions where the variable appears in both the base and exponent, or where the function is a complicated product or quotient ^{[logarithmic-differentiation]}. The method works by taking the natural logarithm of both sides of an equation $y = f(x)$:

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

Then, applying the chain rule to differentiate both sides with respect to $x$:

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

Solving for the derivative:

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

Substituting back the original function for $y$ yields the final derivative. This approach transforms multiplicative relationships into additive ones, simplifying the differentiation process and reducing errors ^{[logarithmic-differentiation]}.

Volume of Solids of Revolution

The volume of a solid obtained by rotating a region bounded by curves $y = f(x)$ and $y = g(x)$ from $x = a$ to $x = b$ about the x-axis is given by ^{[volume-of-solid-of-revolution]}:

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

For rotation about the y-axis, the formula becomes:

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

where $h(y)$ and $k(y)$ are the outer and inner functions respectively. These formulas arise from the disk/washer method and the shell method, both of which slice the solid into thin pieces, compute the volume of each piece, and integrate ^{[volume-of-solid-of-revolution]}.

Worked Examples

Example 1: Testing 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 bound. We evaluate it as a limit:

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

Compute the antiderivative:

$\int \frac{1}{x^2} \, dx = -\frac{1}{x}$

Evaluate the definite integral:

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

Since the limit exists and equals a finite value, the integral converges to 1 ^{[convergence-of-integrals]}.

Example 2: Logarithmic Differentiation

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

Solution:

This function has the variable $x$ in both the base and exponent, making standard differentiation rules awkward. Use logarithmic differentiation.

Take the natural logarithm of both sides:

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

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

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

Solve for $\frac{dy}{dx}$:

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

This result would be difficult to obtain using the power rule or chain rule alone ^{[logarithmic-differentiation]}.

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:

The region is bounded above by $y = \sqrt{x}$ and below by $y = 0$ (the x-axis), from $x = 0$ to $x = 4$. Rotating about the x-axis, we use the disk method:

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

Evaluate:

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

The volume of the solid is $8\pi$ cubic units ^{[volume-of-solid-of-revolution]}.

References

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

AI Disclosure

This article was drafted with the assistance of an AI language model based on personal class notes. The mathematical statements, definitions, and worked examples are derived from the cited notes and standard Calculus II pedagogy. All claims are attributed to source notes via wikilinks. The article has been reviewed for technical accuracy and clarity by a human author.