Application of Inequality in Definite Integration

Definite Integration

Definite integration is one of those ideas that looks abstract at first, but it’s basically a very systematic way of measuring “total change” or “accumulated value” over a fixed interval. Whether you’re tracking distance from speed, total cost from rate, or area under a curve, definite integrals are the mathematical tool that ties it all together.

What Is Definite Integration?

When a function $f(x)$ is defined on a closed interval $[a,b]$, and $F(x)$ is another function such that
$,\frac{d}{dx}(F(x)) = f(x),$ for all $x$ in its domain, then the definite integral is given by:

$ \int_a^b f(x),dx = [F(x) + c]_a^b = F(b) - F(a) $

Here:

• $a$ = lower limit

• $b$ = upper limit

• $F(x)$ = antiderivative of $f(x)$

This represents the net area between the curve $y = f(x)$ and the x-axis from $x = a$ to $x = b$.

Applications of Inequalities in Definite Integration

Inequalities help compare expressions or functions when they’re not equal to each other.
Symbols used: $<, >, \le, \ge, \neq$.

In definite integration, inequalities become powerful tools to:

• estimate the value of an integral

• compare two integrals

• apply average value theorems

• use advanced inequalities like Jensen and Minkowski

Let’s walk through the major results.

Key Inequality Results in Definite Integration

When you're dealing with definite integrals, inequalities act like quick checkpoints—they help you estimate, compare, and bound the value of an integral without fully solving it. This section walks you through the most important inequality results that make definite integration easier.

1. Bounding an Integral Using Function Limits

If a function stays between two constants $m$ and $M$ on $[a,b]$, i.e.,
$m \le f(x) \le M$ for all $x \in [a,b]$, then

$ m(b - a) \le \int_a^b f(x),dx \le M(b - a) $

This is useful for estimating integrals without solving them exactly.

2. Mean Value Theorem for Integrals

If $f(x)$ is continuous on $[a,b]$, then there exists some $c \in [a,b]$ such that:

$ f(c) = \frac{1}{b-a} \int_a^b f(x),dx $

This means the integral equals the area of a rectangle of width $(b-a)$ and height $f(c)$.

3. Comparing Two Functions via Integrals

If $f(x) \ge g(x)$ for all $x \in [a,b]$, then:

$ \int_a^b f(x),dx \ge \int_a^b g(x),dx $

This is extremely common in inequality-based problems.

4. Jensen’s Inequality (Advanced Application)

Let $\phi(x)$ be convex and $f(x)$ be integrated over $[a,b]$. Then:

$ \phi\left(\int_a^b f(x),dx\right) \le \int_a^b \phi(f(x)),dx $

This is widely used in optimization and probability theory.

5. Minkowski’s Inequality (For $L^p$ Spaces)

For functions $f(x)$ and $g(x)$ in $L^p$ space:

$ \left( \int_a^b |f(x) + g(x)|^p,dx \right)^{1/p}
\le
\left( \int_a^b |f(x)|^p,dx \right)^{1/p}
+
\left( \int_a^b |g(x)|^p,dx \right)^{1/p} $

This forms the basis of many results in functional analysis and higher mathematics.

Solved Examples based on Applications of Inequalities in Definite Integration

Example 1: Which of the following is a perfect way to write an integral?
1) $\int_1^3 f(x)$

2) $\int_5^8 f(x) d y$

3) $\int_7^9 f(x) d x$

4) $\int_7^5 f(x) d x$

Solution

As we have learnt,

Definite integration -

When $f(x)$ is integrated in a continuous limit, $(a, b)$: a, and b are known as the limit of integration.

wherein

Where a < x < b

In (A), dx is not written.

In (B), the variables are not the same.

In (D), the lower variable is greater than the upper variable.

Hence, the answer is the option 3.

Example 2: Which of the following is NOT a definite integral?
$
\int_{-5}^5 f(x) d x
$

2) $\int g(y) d y$

$
\int_3 \int_0^5 d u
$

4) $\int_5^6 0 d y$

Solution

As we have learned,

Definite integration -

When $f(x)$ is integrated in a continuous limit, $(a, b)$: a, and b are known as the limit of integration.

wherein

Where$a<x<b$

The upper and lower limits are not specified.

Hence, the answer is the option 2.

Example 3: Which of the following is NOT true?

(1) $\int_a^b f(x) d x=F(b)-F(a)$
$\int_0^0 f(x) d x=-F(a$
$\int_0^a f(x) d x=F(a)-F(b)$
4) $\int_a^a f(x) d x=0$

Solution

As we have learned,

lower and upper limit -

$\int_a^b f(x) d x=(F(x))_a^b$

$=F(b)-F(a)$

wherein

Where a is lower and b is the upper limit.

$\int_0^a f(x) d x=F(0)-F(a)$

and F(0) may or may not be equal to 0

Hence, the answer is the option (2).

Example 4: If $\int_a^b f(x) d x=F(b)-F(a)$; then which of the following is NOT true?

1) "a" s the lower limit

2) "b" is the upper limit

3) $\int_a^b f(x) d x=\int_b^a f(x) d x$

4) If $F(b)=F(a) ; \int_a^b f(x) d x=0$

Solution

As we have learnt,

lower and upper limit -

$\begin{array}{r}\int_a^b f(x) d x=(F(x))_a^b \\ =F(b)-F(a)\end{array}$

wherein

Where a is the lower and b is the upper limit.

$\int_a^b f(x) d x \neq \int_h^a f(x) d x$

Hence, the answer is the option 3.

List of Topics Related to Application of Inequality in Definite Integration

This section provides a comprehensive list of key topics related to the application of inequality in definite integration, offering a structured path for exploring foundational and advanced concepts.

Application of Integrals

Integral of Particular Functions

Indefinite Integrals

Integration by parts

NCERT Resources

This section contain links to essential NCERT resources for Chapter 7 - Integrals, including the chapter notes, solutions, and exemplar solutions for Class 12 Maths.

NCERT Class 12 Maths Notes for Chapter 7 - Integrals

NCERT Class 12 Maths Solutions for Chapter 7 - Integrals

NCERT Class 12 Maths Exemplar Solutions for Chapter 7 - Integrals

Practice Questions based on Application of Inequality in Definite Integration

Below is a link to a dedicated MCQ resource, ideal for testing and sharpening your understanding of how inequalities can be applied to solve definite integration problems in competitive exams like JEE Main.

Application Of Inequality In Definite Integration- Practice Question MCQ

We have provided below the practice questions related to different concepts of integration to improve your understanding: