MAT 266: Calculus for Engineers II

Substitution Method (u-Substitution)

Used to reverse the chain rule. Identify an "inner" function $g(x)$ whose derivative $g'(x)$ is also present.

Integration by Parts

Used to reverse the product rule. A common mnemonic to choose $u$ is LIATE (Logarithmic, Inverse Trig, Algebraic, Trigonometric, Exponential).

Trigonometric Substitution

Used for integrals containing specific forms of square roots.

• For $\sqrt{a^2 - x^2}$, let $x = a\sin\theta$, then $dx = a\cos\theta d\theta$. Identity: $1 - \sin^2\theta = \cos^2\theta$.
• For $\sqrt{a^2 + x^2}$, let $x = a\tan\theta$, then $dx = a\sec^2\theta d\theta$. Identity: $1 + \tan^2\theta = \sec^2\theta$.
• For $\sqrt{x^2 - a^2}$, let $x = a\sec\theta$, then $dx = a\sec\theta\tan\theta d\theta$. Identity: $\sec^2\theta - 1 = \tan^2\theta$.

Integration by Partial Fractions

Used to integrate rational functions $\frac{P(x)}{Q(x)}$ by decomposing them into simpler, integrable fractions based on the factors of the denominator $Q(x)$.

Numerical Integration (Approximation)

For a definite integral $\int_a^b f(x)dx$ with $\Delta x = \frac{b-a}{n}$.

• Midpoint Rule: $$ M_n = \Delta x [f(\bar{x}_1) + f(\bar{x}_2) + \dots + f(\bar{x}_n)] $$
• Trapezoidal Rule: $$ T_n = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + \dots + 2f(x_{n-1}) + f(x_n)] $$
• Simpson's Rule (n must be even): $$ S_n = \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n)] $$

Improper Integrals

Integrals over an infinite interval or where the function has an infinite discontinuity.

• Type 1 (Infinite Interval): $$ \int_a^\infty f(x)dx = \lim_{t \to \infty} \int_a^t f(x)dx $$
• Type 2 (Discontinuity at b): $$ \int_a^b f(x)dx = \lim_{t \to b^-} \int_a^t f(x)dx $$

Area Between Curves

If $f(x) \ge g(x)$ on $[a, b]$:

Volumes of Solids of Revolution

• Disk Method (Rotation about x-axis): $$ V = \pi \int_a^b [R(x)]^2 dx $$
• Washer Method (Rotation about x-axis): $$ V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2) dx $$
• Cylindrical Shell Method (Rotation about y-axis): $$ V = 2\pi \int_a^b x h(x) dx $$

Arc Length

Work

Work done by a variable force $F(x)$ from $x=a$ to $x=b$.

• Springs (Hooke's Law): $F(x) = kx$. Work to stretch from $a$ to $b$ is $\int_a^b kx \, dx$.
• Pumping Liquid: $W = \int_a^b \rho g A(y) D(y) dy$, where $\rho$ is density, $g$ is gravity, $A(y)$ is the area of a slice, and $D(y)$ is the distance it's moved.

Sequences

A sequence $\{a_n\}$ converges to a limit $L$ if $\lim_{n \to \infty} a_n = L$. Otherwise, it diverges.

Series

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

• Geometric Series: $\sum_{n=1}^\infty ar^{n-1}$ converges to $\frac{a}{1-r}$ if $|r| < 1$. It diverges if $|r| \ge 1$.
• Test for Divergence: If $\lim_{n \to \infty} a_n \neq 0$ or the limit does not exist, then the series $\sum a_n$ diverges.

Power Series

A series of the form $\sum_{n=0}^\infty c_n (x-a)^n$. We find its Radius of Convergence (R) and Interval of Convergence (I), often using the Ratio Test.

Taylor and Maclaurin Series

Representing a function as an infinite polynomial.

• Taylor Series centered at $a$: $$ f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n $$
• Maclaurin Series (centered at $a=0$): $$ f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n $$

You can differentiate and integrate known power series term-by-term to create new series.

Parametric Equations

A curve is defined by $x=f(t), y=g(t)$.

• Tangent Line Slope: $$ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} $$
• Arc Length: $$ L = \int_\alpha^\beta \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt $$

Polar Coordinates

Points are defined by $(r, \theta)$. Conversions: $x = r\cos\theta$, $y = r\sin\theta$, and $r^2 = x^2 + y^2$.

• Area enclosed by a polar curve $r=f(\theta)$ from $\theta=\alpha$ to $\theta=\beta$:
  $$ A = \frac{1}{2} \int_\alpha^\beta [r(\theta)]^2 d\theta $$

Symbol	Name / Meaning	Summary & Context
$\int$	Integral Sign	The fundamental symbol for integration, representing an infinite sum of infinitesimally small quantities.
$\int_a^b$	Definite Integral	Represents the net area under the curve of a function from point $a$ to point $b$.
$dx$	Differential	Represents an infinitesimally small change in the variable $x$. Indicates the variable of integration.
$f'(x)$, $\frac{dy}{dx}$	Derivative	Represents the instantaneous rate of change of a function. Crucial for arc length and parametric slope.
$\sum$	Sigma / Summation	Represents the sum of a sequence of terms. Used extensively in series and numerical integration.
$\lim_{x \to c}$	Limit	Describes the value that a function or sequence "approaches" as the input approaches some value. Essential for improper integrals and sequences/series.
$\infty$	Infinity	A concept representing a quantity without bound. Used in limits for improper integrals and convergence of sequences.
$\Delta x$	Delta x	Represents a finite change in $x$. In numerical integration, it's the width of each subinterval: $\frac{b-a}{n}$.
$\{a_n\}$	Sequence	An ordered list of numbers, e.g., $a_1, a_2, a_3, \dots$. We test sequences for convergence or divergence.
$S_n$	Partial Sum	The sum of the first $n$ terms of a series. The limit of $S_n$ as $n \to \infty$ determines the convergence of the series. Also used for Simpson's Rule approximation.
$n!$	Factorial	The product of all positive integers up to $n$. Key component of Taylor and Maclaurin series formulas.
$(r, \theta)$	Polar Coordinates	A way to define points in a plane using a distance from the origin ($r$) and an angle ($\theta$).
$(\alpha, \beta)$	Alpha, Beta	Often used as the start and end angles for integration in polar coordinates, or start and end time in parametric equations.