Chapter 5 Definition of a Definite Integral

Let’s start with the book’s definition of a definite integral, and then I’ll explain in English.

Definition:

If f is a continuous function defined for the interval [a, b], we divide this interval into n subintervals of equal width such that ∆X = (b – a) / (n). We let x₀ (= a), x₁, x₂,…x_n (= b) be the endpoints of these subintervals and we choose sample points x₁^*, x₂^*,… x_n^* in these subintervals, so x_i^* lies in the ith subinterval [x_i-1, x_i]. Then the definite integral of f from a to b is

All of that means that the area under the graph is equal to Lim_{n→ ∞} ∆X {}

As we defined in the last section.

This symbol indicates that we are finding the area under a curve. When we are calculating the area under a curve we are solving an integral. So this symbol is the integral symbol.

The a and b tell us the interval on which we are finding the area under the curve. a stands for the lower limit, and b stands for the upper limit.