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 $$\displaystyle \int_a^b f(x) \;d x = – \int_b^a f(x) \;d x$$
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
$$\int_a^b f(x) \; dx \leq \int_a^b g(x) \; dx.$$