5.3 | Riemann Sums and Definite Integrals


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

  1. If \(f\) is defined at \(x = a\), then \(\displaystyle \int_a^a f(x) \;d x = 0\)
  2. 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

  1. \(\displaystyle \int_a^b k f(x) \; dx = k \int_a^b f(x) \; dx \)
  2. \(\displaystyle \int_a^b f(x) \pm g(x) \; dx = \int_a^b f(x) \; dx \pm \int_a^b g(x) \; dx \)
Preservation of Inequality

  1. If \(f\) is integrable and nonnegative on the closed interval \([a,b]\), then
    $$
    0 \leq \int_a^b f(x) \; dx.
    $$
  2. 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
    $$
    \int_a^b f(x) \; dx \leq \int_a^b g(x) \; dx.
    $$

E 5.3 Exercises