- 5.1 | Antiderivatives and Indefinite Integration
- 5.2 | Area
- 5.4 | The Fundamental Theorem of Calculus
- 5.5 | Integration by Substitution

**Definition of Riemann Sum**

Let \(f\) be defined on the closed interval \([a,b]\), and let \(\Delta\) be a partition of \([a,b]\) given by

$$

a = x_0 < x_1 < \cdots < x_{n-1} < x_n = b $$ where \(\Delta x_i\) is the width of the \(i\)th subinterval $$ [x_{i-1}, x_i] $$ If \(c_i\) is any point in the \(i\)th subinterval, then the sum $$ \sum_{i=1}^n f(c_i) \Delta x_i, \quad x_{i-1} \leq c_i \leq x_i $$ is called a

**Riemann sum**of \(f\) for the partition \(\Delta\).

**Definition of Definite Integral**

If \(f\) is defined on the closed interval \([a,b]\) and the limit of Riemann sums over partitions \(\Delta\)

$$

\lim_{||\Delta || \rightarrow 0} \sum_{i=1}^n f(c_i) \Delta x_i

$$

exists, then \(f\) is said to be integrable on \([a,b]\) and the limit is denoted

$$

\lim_{||\Delta || \rightarrow 0} \sum_{i=1}^n f(c_i) \Delta x_i = \int_a^b f(x) \; dx.

$$

The limit is called the

**definite integral**of \(f\) from \(a\) to \(b\). The number \(a\) is the

**lower limit**of integration, and the number \(b\) is called the

**upper limit**of integration.

**Continuity Implies Integrability**

If a function \(f\) is continuous on the closed interval \([a,b]\), then \(f\) is integrable on \([a,b]\). That is, \(\int_a^b f(x) \; dx\) exists.

**The Definite Integral as the Area of a Region**

If \(f\) is continuous and nonnegative on the closed interval \([a,b]\), then the area of the region bounded by the graph of \(f\), the \(x\)-axis, and the vertical lines \(x=a\) and \(x =b\) is

$$

\text{Area} = \int_a^b f(x) \; dx.

$$

**Definitions of Two Special Definite Integrals**

- If \(f\) is defined at \(x = a\), then \(\displaystyle \int_a^a f(x) \;d x = 0\)
- If \(f\) is integrable \([a,b]\), then \(\displaystyle \int_a^b f(x) \;d x = – \int_b^a f(x) \;d x\)

**Additive Interval Property**

If \(f\) is integrable on the three closed intervals determined by \(a\), \(b\), and \(c\), then

$$

\int_a^b f(x) \; dx = \int_a^c f(x) \; dx + \int_c^b f(x) \; dx

$$

**Properties of Definite Integrals**

If \(f\) and \(g\) are integrable on \([a,b]\) and \(k\) is a constant, then the functions \(kf\) and \(f \pm g\) are integrable on \([a,b]\), and

- \(\displaystyle \int_a^b k f(x) \; dx = k \int_a^b f(x) \; dx \)
- \(\displaystyle \int_a^b f(x) \pm g(x) \; dx = \int_a^b f(x) \; dx \pm \int_a^b g(x) \; dx \)

**Preservation of Inequality**

- If \(f\) is integrable and nonnegative on the closed interval \([a,b]\), then

$$

0 \leq \int_a^b f(x) \; dx.

$$ - If \(f\) is integrable and nonnegative on the closed interval \([a,b]\) and \(f(x) \leq g(x)\) for every \(x\) in \([a,b]\), then

$$

\int_a^b f(x) \; dx \leq \int_a^b g(x) \; dx.

$$