Calculus II Final Exam Review: Key Topics and Concepts

Definite and Indefinite Integrals

Evaluating Integrals

Integration is a fundamental concept in calculus, used to find areas, accumulated quantities, and solve differential equations. There are two main types:

• Indefinite Integrals: Represent the family of all antiderivatives of a function. The general form is $\int f(x)\,dx = F(x) + C$, where $C$ is the constant of integration.

• Definite Integrals: Represent the net area under a curve between two points. The general form is $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$.

Example: $\int_0^2 x^2\,dx = \left.\frac{x^3}{3}\right|_0^2 = \frac{8}{3}$

Area Between Curves

Finding the Area

The area between two curves $y = f(x)$ and $y = g(x)$ from $x = a$ to $x = b$ is given by:

• Formula: $\int_a^b [f(x) - g(x)]\,dx$, where $f(x) \geq g(x)$ on $[a, b]$.

Example: Find the area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$:

• $\int_0^1 (x - x^2)\,dx = \left.\frac{x^2}{2} - \frac{x^3}{3}\right|_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$

Improper Integrals

Evaluating Improper Integrals

Improper integrals involve infinite limits or unbounded integrands. They are evaluated using limits:

• Infinite Interval: $\int_a^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx$

• Unbounded Integrand: $\int_a^b f(x)\,dx$ where $f(x)$ is unbounded at $a$ or $b$ is defined as a limit approaching the point of discontinuity.

Example: $\int_1^{\infty} \frac{1}{x^2}\,dx = \lim_{b \to \infty} \left(-\frac{1}{x}\right)\Big|_1^b = 1$

Volumes of Solids of Revolution

Disk, Washer, and Shell Methods

These methods are used to find the volume of a solid generated by revolving a region around an axis.

• Disk Method: $V = \pi \int_a^b [R(x)]^2\,dx$

• Washer Method: $V = \pi \int_a^b ([R_{\text{outer}}(x)]^2 - [R_{\text{inner}}(x)]^2)\,dx$

• Shell Method: $V = 2\pi \int_a^b r(x) h(x)\,dx$

Example: Volume of solid formed by revolving $y = x$ from $x = 0$ to $x = 1$ about the y-axis using the shell method:

• $V = 2\pi \int_0^1 x \cdot x\,dx = 2\pi \int_0^1 x^2\,dx = 2\pi \left.\frac{x^3}{3}\right|_0^1 = \frac{2\pi}{3}$

Work

Calculating Work Using Integrals

Work is the product of force and distance. When force varies, work is calculated as:

• Formula: $W = \int_a^b F(x)\,dx$

Example: Lifting a 10-kg object 5 meters: $W = mgd = 10 \times 9.8 \times 5 = 490$ Joules

Additional info: For variable force, such as stretching a spring, use Hooke's Law: $F(x) = kx$.

Sequences and Series

Limits of a Sequence

A sequence $\{a_n\}$ converges to $L$ if $\lim_{n \to \infty} a_n = L$.

• Example: $a_n = \frac{1}{n}$ converges to 0 as $n \to \infty$.

Convergence and Divergence of Infinite Series

An infinite series $\sum_{n=1}^{\infty} a_n$ converges if the sequence of partial sums converges.

• Common Tests:

  • n-th Term Test: If $\lim_{n \to \infty} a_n \neq 0$, the series diverges.

  • Geometric Series Test: $\sum ar^n$ converges if $|r| < 1$.

  • p-Series Test: $\sum \frac{1}{n^p}$ converges if $p > 1$.

  • Comparison, Ratio, Root, and Integral Tests are also commonly used.

Example: $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges (p-series with $p=2$).

Sum of an Infinite Series

Some series have a closed-form sum.

• Geometric Series: $\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r}$ for $|r| < 1$

Example: $\sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n = \frac{1}{1 - 1/2} = 2$

Power Series: Radius and Interval of Convergence

A power series is $\sum_{n=0}^{\infty} c_n (x - a)^n$. The radius of convergence $R$ is found using the ratio or root test.

• Ratio Test: $R = \lim_{n \to \infty} \left| \frac{c_n}{c_{n+1}} \right|$

• Interval of Convergence: $|x - a| < R$

Example: $\sum_{n=0}^{\infty} x^n$ converges for $|x| < 1$

Taylor and Maclaurin Series

Definitions and Formulas

A Taylor series for $f(x)$ centered at $a$ is:

• $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$

• A Maclaurin series is a Taylor series centered at $a = 0$.

Example: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

Calculus with Taylor and Maclaurin Series

Series can be differentiated and integrated term by term within their interval of convergence.

• Differentiation: $\frac{d}{dx} \left( \sum_{n=0}^{\infty} c_n (x - a)^n \right ) = \sum_{n=1}^{\infty} n c_n (x - a)^{n-1}$

• Integration: $\int \sum_{n=0}^{\infty} c_n (x - a)^n dx = \sum_{n=0}^{\infty} \frac{c_n}{n+1} (x - a)^{n+1} + C$

Parametric Equations

Calculus with Parametric Equations

Parametric equations express $x$ and $y$ as functions of a parameter $t$.

• Derivative: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

• Arc Length: $L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$

Example: For $x = t^2$, $y = t^3$, $\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2}$

Polar Coordinates

Calculus with Polar Equations

Polar coordinates represent points as $(r, \theta)$, where $r$ is the radius and $\theta$ is the angle.

• Area: $A = \frac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^2 d\theta$

• Arc Length: $L = \int_{\alpha}^{\beta} \sqrt{[r(\theta)]^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta$

Example: Area inside $r = 2\sin\theta$ from $\theta = 0$ to $\pi$:

• $A = \frac{1}{2} \int_0^{\pi} [2\sin\theta]^2 d\theta = 2 \int_0^{\pi} \sin^2\theta d\theta$

Summary Table: Key Tests for Series Convergence

Test	When to Use	Convergence Criteria
n-th Term Test	Any series	If $\lim_{n \to \infty} a_n \neq 0$, diverges
Geometric Series	$a r^n$ form	Converges if $|r| < 1$
p-Series	$1/n^p$ form	Converges if $p > 1$
Ratio Test	Factorials, exponentials	Converges if $L < 1$
Root Test	Powers of $n$	Converges if $L < 1$
Comparison Test	Positive term series	Compare to known series

Additional info: For more practice, refer to the textbook exercises listed for each chapter.