Definite integrals as limit of a sum

Definite Integrals as Limit of a Sum A definite integral is the area bounded by the curve of a function and the vertical line segments connecting the points...

Definite Integrals as Limit of a Sum

A definite integral is the area bounded by the curve of a function and the vertical line segments connecting the points on the graph.

The definite integral as a limit of a sum is a way to compute the area of the bounded region by breaking it up into smaller, equal parts. The idea is to divide the interval of integration into a sequence of subintervals of equal width, and then take the limit of the areas of these subintervals as the width of the subintervals approaches zero.

Formally, the definite integral as a limit of a sum is defined as:

$\int_a^b f(x) dx \approx \sum_{i=1}^n f(c_i) \Delta x$

where:

(a) and (b) are the endpoints of the interval of integration.
(n) is the number of subintervals.
(c_i) is the midpoint of the interval interval.
(\Delta x = \frac{b-a}{n}) is the width of each subinterval.

Intuitively, this definition says that the definite integral represents the limit of the sum of the areas of the rectangles with side lengths (\Delta x) that cover the interval ([a, b]).

Examples:

If (f(x) = x^2), then the definite integral from (a=0) to (b=1) is (1), which is the area of the unit square.

$\int_0^1 x^2 dx = \sum_{i=1}^n (c_i)^2 \Delta x = \frac{1}{n} \sum_{i=1}^n i^2 = 1$

If (f(x) = \frac{1}{x}), then the definite integral from (a=1) to (b=e) is (\infty), since the function blows up as (x\to 0).

$\int_1^e \frac{1}{x} dx = \lim_{n\to \infty} \sum_{i=1}^n \frac{1}{c_i} \Delta x = \infty$

The limit of the sum of the areas of these rectangles as the width of the subintervals approaches zero is equal to the definite integral, which represents the exact area of the original bounded region.