Definite Integrals as the Limit of a Sum

Welcome to mastering definite integrals as the limit of a sum! This concept is fundamental in calculus, bridging the gap between approximating areas with rectangles and precisely calculating them. It's a key topic for competitive exams like JEE Mathematics.

Understanding the Core Idea

Imagine you want to find the area under a curve, say $f(x)$ , between two points $a$ and $b$ on the x-axis. The idea of the limit of a sum is to approximate this area by dividing the region into many thin rectangles and then letting the width of these rectangles approach zero, making the approximation infinitely accurate.

The area under a curve can be found by summing the areas of infinitely many infinitesimally thin rectangles.

We divide the interval $[a, b]$ into $n$ equal subintervals. Each subinterval has a width $\Delta x = (b-a)/n$ . We then form rectangles using the function's value at a point within each subinterval (e.g., the right endpoint) and sum their areas. As $n$ approaches infinity, this sum converges to the exact area.

Let the interval be $[a, b]$ . We divide this into $n$ equal subintervals, each of width $\Delta x = \frac{b-a}{n}$ . The endpoints of these subintervals are $x_0=a, x_1=a+\Delta x, x_2=a+2\Delta x, \dots, x_n=a+n\Delta x=b$ . For each subinterval $[x_{i-1}, x_i]$ , we choose a sample point $c_i$ . The area of the rectangle over this subinterval is $f(c_i) \Delta x$ . The sum of the areas of these $n$ rectangles is $S_n = \sum_{i=1}^{n} f(c_i) \Delta x$ . The definite integral of $f(x)$ from $a$ to $b$ , denoted as $\int_a^b f(x) dx$ , is defined as the limit of this sum as $n o \infty$ : $\int_a^b f(x) dx = \lim_{n o \infty} \sum_{i=1}^{n} f(c_i) \Delta x$ .

Choosing Sample Points

The choice of the sample point $c_i$ within each subinterval affects the sum. For competitive exams, common choices are the left endpoint, the right endpoint, or the midpoint of the subinterval.

Sample Point	Formula for $c_i$	Summation Form
Left Endpoint	$c_i = a + (i-1)\Delta x$	$\lim_{n o \infty} \sum_{i=1}^{n} f(a + (i-1)\Delta x) \Delta x$
Right Endpoint	$c_i = a + i\Delta x$	$\lim_{n o \infty} \sum_{i=1}^{n} f(a + i\Delta x) \Delta x$
Midpoint	$c_i = a + (i - \frac{1}{2})\Delta x$	$\lim_{n o \infty} \sum_{i=1}^{n} f(a + (i - \frac{1}{2})\Delta x) \Delta x$

Key Formulas and Steps

To solve problems involving definite integrals as a limit of a sum, follow these steps:

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Let's break down the steps:

Identify $f(x)$ , $a$ , and $b$ : From the given problem, determine the function $f(x)$ and the interval $[a, b]$ .
Calculate $\Delta x$ : $\Delta x = \frac{b-a}{n}$ .
Choose Sample Point $c_i$ : Typically, $c_i = a + i\Delta x$ (right endpoint) is the most convenient for many problems.
Form the Summation: Substitute $f(c_i)$ and $\Delta x$ into the summation formula: $\sum_{i=1}^{n} f(c_i) \Delta x$ .
Evaluate the Limit: Calculate $\lim_{n o \infty} \sum_{i=1}^{n} f(c_i) \Delta x$ . This often involves using standard summation formulas for powers of $i$ (e.g., $\sum i$ , $\sum i^2$ , $\sum i^3$ ) and then evaluating the limit of the resulting polynomial in $n$ .

Remember the standard summation formulas: $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$ , $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$ , $\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2} ight)^2$ . These are crucial for simplifying the summation before taking the limit.

Example: Area under $f(x) = x^2$ from 0 to 2

Let's find the area under $f(x) = x^2$ from $a=0$ to $b=2$ using the limit of a sum with right endpoints.

Identify: $f(x) = x^2$ , $a=0$ , $b=2$ .
$\Delta x$ : $\Delta x = \frac{2-0}{n} = \frac{2}{n}$ .
$c_i$ (Right Endpoint): $c_i = a + i\Delta x = 0 + i\left(\frac{2}{n} ight) = \frac{2i}{n}$ .
Summation: $f(c_i) = \left(\frac{2i}{n} ight)^2 = \frac{4i^2}{n^2}$ . The sum is $\sum_{i=1}^{n} f(c_i) \Delta x = \sum_{i=1}^{n} \left(\frac{4i^2}{n^2} ight) \left(\frac{2}{n} ight) = \sum_{i=1}^{n} \frac{8i^2}{n^3}$ .
Limit: $\lim_{n o \infty} \sum_{i=1}^{n} \frac{8i^2}{n^3} = \lim_{n o \infty} \frac{8}{n^3} \sum_{i=1}^{n} i^2$ . Using the formula for $\sum i^2$ : $\lim_{n o \infty} \frac{8}{n^3} \left(\frac{n(n+1)(2n+1)}{6} ight)$ . Now, simplify the expression inside the limit: $\frac{8}{n^3} \frac{2n^3 + 3n^2 + n}{6} = \frac{8(2n^3 + 3n^2 + n)}{6n^3} = \frac{16n^3 + 24n^2 + 8n}{6n^3}$ . To evaluate the limit as $n o \infty$ , we look at the ratio of the leading coefficients: $\frac{16}{6} = \frac{8}{3}$ . Therefore, the area is $\frac{8}{3}$ .

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Connection to Fundamental Theorem of Calculus

While the limit of a sum is the definition of the definite integral, the Fundamental Theorem of Calculus provides a much more efficient way to evaluate definite integrals. Understanding the limit of a sum helps appreciate why the FTC works. The FTC states that if $F'(x) = f(x)$ , then $\int_a^b f(x) dx = F(b) - F(a)$ .

What is the primary purpose of using the limit of a sum to define a definite integral?

To approximate the area under a curve using an infinite number of infinitesimally thin rectangles, leading to an exact calculation.

What is the formula for

\Delta x

when dividing an interval

[a, b]

into

n

subintervals?

$\Delta x = \frac{b-a}{n}$

Learning Resources

Definite Integrals as Limits of Riemann Sums - Khan Academy(video)

This video provides a clear explanation of Riemann sums using left and right endpoints, which is foundational to understanding definite integrals as limits of sums.

Definite Integral as a Limit of a Sum - Paul's Online Math Notes(documentation)

A comprehensive guide covering the definition of the definite integral as a limit of a sum, including examples and key formulas.

The Definite Integral - MIT OpenCourseware(video)

This lecture from MIT OCW delves into the concept of the definite integral and its relationship to areas, using the limit of sums.

Riemann Sums and Definite Integrals - Brilliant.org(blog)

An interactive explanation of Riemann sums and how they lead to the definition of the definite integral, with visual aids.

Definite Integral as Limit of Sum - YouTube Tutorial(video)

A step-by-step video tutorial demonstrating how to calculate definite integrals using the limit of a sum method.

Calculus: Definite Integrals as Limits of Riemann Sums(documentation)

This section from LibreTexts provides a detailed explanation of the definite integral as a limit of Riemann sums, including formal definitions and examples.

Understanding the Definite Integral - Mathematics Stack Exchange(blog)

A forum discussion offering various perspectives and explanations on the concept of the definite integral and its summation definition.

Definite Integral - Wikipedia(wikipedia)

The Wikipedia page on definite integrals, which includes a section on its definition as a limit of a sum (Riemann integral).

Calculus I: Definite Integrals as Limits of Sums - University of Houston(documentation)

Lecture notes that clearly outline the process of evaluating definite integrals using the limit of a sum, with practice problems.

The Geometric Interpretation of the Definite Integral(blog)

This discussion focuses on the geometric meaning of the definite integral as the area under a curve, reinforcing the limit of sums concept.

Definite Integrals as Limit of a Sum