Definite Integral as Limit of a Sum
Assuming that ƒ is a continuous function and positive on the interval [a, b]. So, its graph is above the x-axis.
Definite integral is the area bounded by the curve y = f(x), the ordinates x = a and x = b and x-axis.
Now to evaluate this area, consider the region ABCD in the figure below,
Let x0 = a and xn = b.
Now divide the interval [a, b] into n equal subintervals denoted by [x0, x1], [x1, x2], [x2, x3] ….[xr-1, xr] …..[xn-1, xn]
where x0 = a, x1 = a + h, x2 = a + 2h …. and xn = a + nh or n =. As n ⇢ ∞, h ⇢ 0.
The region ABCD under consideration is the sum of n subregions, where each subregion is defined on subintervals [xr – 1, xr], where, r = 1, 2, 3, …, n.
It can be seen in the above figure that, now the area of the triangle POFR is calculated as,
A = PQ × PR
⇒ A = (xr– xr–1) × f(xr-1)
As xr– xr–1 → 0, i.e., h → 0, the area above becomes a nearly perfect rectangle. Now the area under the curve can be broken into n different rectangles adding all these rectangles’ areas we get the area under the curve.
sn and Sn denote the sum of areas of all lower rectangles and upper rectangles raised over subintervals [xr-1, xr] for r = 1, 2, 3,…. respectively.
As n → ∞ strips become narrower and narrower, so, the limiting values of (2) and (3) are the same in both cases and the common limiting value is the required area under the curve.
So,
Now, this equation can also be re-written as,
where,
This expression is knows as definition of definite integral as limit of sum.
Example: Find as the limit of sum.
Solution:
By the definition given above,
Where
Here, a = 0 and b =2, f(x) = x2 + 1, h =
Definite Integral | Definition, Formula & How to Calculate
A definite integral is an integral that calculates a fixed value for the area under a curve between two specified limits. The resulting value represents the sum of all infinitesimal quantities within these boundaries. i.e. if we integrate any function within a fixed interval it is called a Definite Integral. The starting point of the interval is the lower limit of the definite integral whereas the endpoint of the interval is the upper limit of the definite integral.
The definite integral is widely used in advanced mathematics or mechanics or others. It is used for calculating the area of irregular curves, and the volume of random shapes, and there are many other advantages.
In this article, we will learn about, Definite Integrals, formulas, applications, and others in detail.
Table of Content
- What is a Definite Integral?
- Definite Integral Definition
- Definite Integral Formula
- Evaluating Definite Integrals
- Definite Integral as Limit of a Sum
- Definite Integral and Fundamental Theorems of Calculus
- First Fundamental Theorem of Calculus
- Second Fundamental Theorem of Calculus
- Steps for Calculating Definite Integral
- Properties of Definite Integral
- Definite Integral by Parts
- Applications of Definite Integral
- Definite Integral Examples
- Definite Integral Practice Problems