Integral limit definition

Question

No limit is visible in integral notation, but integration is defined in terms of a limit.  Give the definition; explain what is being limited, and how this allows an integral to calculate the exact area under a curve.

Answer 1

A Riemann Sum for ∫_a^b f(x) dx is when you chop up the interval [a,b] into n subintervals [a_i,a_i+1] where a = a₀ < a₁ < ... < a_n = b and pick an x_i in each interval. You get rectangles of width a_i+1 - a_i and height f(x_i). The sum of their areas approximates the integral.

 

To get the exact value of the integral from the Riemann Sum, you take the limit as follows:

 

1. n → ∞ (Large number of subintervals).

 

2. max_i=1,...n(a_i+1 - a_i) → 0 (All subintervals have small length).

 

This method of computing the integral is called Riemann Integration, and is used as the definition for Integration in Calculus 1 and 2, since it is sufficient for integrals of continuous functions to be well defined.