Week 01-1: Math 1A Review

Key Concepts from Chapter 5

Section 5.1

Area Under a Curve

The area \(a\) of the region \(S\) that lies under the graph of the continuous function \(f\) is

\begin{align} A &= \lim_{n \to \inf} [ f(x_1)\Delta x + f(x_2)\Delta x + \ldots + f(x_n)\Delta x ] \\ &= \lim_{n \to \inf} \Delta x \sum_{i=1}^{n} f(x_i) \end{align}

Distance Traveled

\begin{align} d = \lim_{n \to \inf} \sum_{i=1}^{n} \Delta t f(t_i) \end{align}

Section 5.2

Definite Integral

\begin{align} \int_{a}^{b} f(x) dx = \lim_{n \to \inf} \sum_{i=1}^{n} f(x_i) \Delta x \end{align}

Midpoint Rule

\begin{align} \int_{a}^{b} f(x) dx \approx \sum_{i=1}^{n} f(\underbrace{\bar{x}_i}_{\text{Midpoint Value}}) \Delta x \end{align}

Section 5.3

Fundamental Theorem of Calculus (Part 1)

If \(f\) is continuous on \([a, b]\), then the function \(g\) defined by

\begin{align} g(x) = \int_{x}^{a} f(t) dt \quad a \leq x \leq b \end{align}

is continuous on \([a, b]\) and differentiable on \((a, b)\), and \(g'(x) = f(x)\).

Fundamental Theorem of Calculus (Part 2)

If \(f\) is continuous on \([a, b]\), then

\begin{align} \int_{a}^{b} f(x) dx = F(b) - F(a) \end{align}

where \(F\) is any antiderivative of \(f\), that is, a function such that \(F'=f\).

Section 5.4

Antiderivative Reference

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C \enspace (n \ne -1)$$	$$\int \frac{1}{x} dx = \ln |x| + C$$
$$\int e^x dx = e^x + C$$	$$\int b^x dx = \frac{b^x}{\ln b} + C$$
$$\int \sin x dx = -\cos x + C$$	$$\int \cos x dx = \sin x + C$$
$$\int \sec^2 x dx = \tan x + C$$	$$\int \csc^2 x dx = -\cot x + C$$
$$\int \sec x \tan x dx = \sec x + C$$	$$\int \csc x \cot x dx = -\csc x + C$$
$$\int \sinh x dx = \cosh x + C$$	$$\int \cosh x dx = \sinh x + C$$
$$\int \tan x dx = \ln |\sec x| + C$$	$$\int \cot x dx = \ln |\sin x| + C$$
$$\int \frac{1}{x^2+a^2} dx = \frac{1}{a} \arctan{\frac{x}{a}} + C$$	$$\int \frac{1}{\sqrt{a^2-x^2}} dx = \arcsin{\frac{x}{a}} + C,\enspace a > 0$$

Net Change Theorem

\begin{align} \int_{a}^{b} F'(x) dx = F(b) - F(a) \end{align}

Section 5.5

The Substitution Rule

If \(u=g(x)\) is a differentiable function whose range is an interval \(I\) and \(f\) is continuous at \(I\), then:

\begin{align} \int f(g(x))g'(x) dx = \int f(u) du \end{align}

Tips for choosing \(u\)

• A function whose derivative appears in the integrand.
• A function that is the inner function of a composite function.
• A function raised to the highest power.
• A function in the denominator.
• A function that is under a radical.