Limit of Integration

Last Updated : 02 Feb, 2024

In integration, the limits are the numbers that set the range for calculating the definite integral of a function. The definite integral, ∫f(x)dx, involves finding the antiderivative F(x) and then evaluating it at the upper and lower limits, [a, b].

In this article, we have covered the basic concept of integration, formulas for limits of integration, the meaning of integration, how to change limits, how to find limits, and the application of limits of integration. At the end of this article, you will learn bout the limits of integration by the solved examples provided and practice questions to test the learning for yourself.

Table of Content

What is Integration?
What are the Limits of Integration?
How to Change the Limits of Integration?
Formulas of Limits of Integration
Application of Limits of Integration

What is Integration?

Integration is a mathematical concept used to find the accumulation or total of a quantity, often represented by a function, over a specified interval. It involves finding the antiderivative (or indefinite integral) of a function.

In simpler terms, integration helps calculate the area under a curve on a graph or the net change in a quantity. There are two main types of integration: indefinite integration, which results in a function with an arbitrary constant, and definite integration, which calculates a specific numerical value by determining the accumulated quantity within specified upper and lower limits.

Types of Integration

There are two types of integration:

Definite Integral
Indefinite Integral

Let’s discuss these in detail as follows.

Definite Integral

Definite integral has both upper and lower limits. It’s like looking at a specific range on the number line, where the variable x is limited. People also call it the Riemann Integral.
Representes as:
$\int_{a}^{b} f(x) \,dx$
Here, a and b are the limits.

Indefinite Integral

Indefinite integrals don’t have upper and lower limits. They are represented like this:
$\int f(x) \,dx = F(x) + C$
In this formula, C is any constant, and f(x) is the function being integrated.

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What are the Limits of Integration?

The limits of integration are the numbers we use to set the range for integrating a function. When we integrate a function, which is like finding the opposite of differentiation, we get what’s called an antiderivative.

To figure out the definite integral between two points, say [a, b], we subtract the antiderivative’s values at (b) from its value at (a). In this range, (a) is the upper limit, and (b) is the lower limit.

$\int_{a}^{b} f(x) \,dx = [F(x)]_{a}^{b} = F(a) - F(b)$

This process helps us find the area under the curve between two points. It’s like calculating the space enclosed by the curve. When we use these limits in integration, it’s called a definite integral, and the result is a specific number. Unlike indefinite integrals, definite integrals don’t have a constant term in the final answer.

Upper Limits of Integration

The upper limit of integration refers to the higher endpoint in a specified range when calculating a definite integral. In the context of integration, particularly in ∫[a, b] f(x) dx, the upper limit is represented by ‘b’.

This signifies the value at which the integration process concludes. When evaluating the integral between ‘a’ and ‘b’, we find the antiderivative of the function and subtract the value of this antiderivative at ‘a’ from its value at ‘b’.

Lower Limits of Integration

The lower limit of integration is one of the numbers that defines the range for calculating the definite integral of a function. When we perform definite integration ∫[a, b] f(x) dx, where a and b are limits, we find the antiderivative F(x) and then evaluate it at the upper limit (F(b)) and subtract the value at the lower limit (F(a)).

In this context, the number ‘a’ is referred to as the lower limit, marking the starting point of the interval, while ‘b’ is the upper limit, representing the endpoint.

Steps to Find the Limits of Integration

To find the limits of integration, we can use the following steps for any integral.

First, we solve the integration problem by figuring out the antiderivative of a function, represented as ∫_a^bf(x).dx = [F(x)]_a^b.
The second step is applying the limits [a, b] to the antiderivative.
- This means substituting the values of a and b into the antiderivative. The final result is obtained by subtracting F(a) from F(b), expressed as [F(x)] evaluated from a to b i.e., F(a) − F(b).

In simple terms, the limits of integration help us find the specific numerical value of the given integral expression.

How to Find Upper and Lower Limit of Integration

If you are given a definite integral like ∫_a^b f(x) dx, where f(x) is the function and a and b are the limits of integration, then:

a is the lower limit of the integration, and
b is the upper limit of the integration.

How to Change the Limits of Integration?

Changing the limits of integration is a process that involves a few simple steps:

Identify Original Limits: Pinpoint the initial integration limits, commonly labeled as ‘a’ and ‘b.’ These values indicate the starting and ending points of integration.

Determine New Limits: Choose new limits, typically represented as ‘c’ and ‘d.’ ‘c’ becomes the updated lower limit, while ‘d’ becomes the new upper limit.

Adjust the Integral: Subtract ‘a’ from both upper and lower limits, and also subtract ‘c’ from both. This adjustment ensures the integral remains consistent.

Add the Difference to the Integral: The subtraction in the previous step yields a difference, referred to as ‘k.’ Add ‘k’ to the integral to accommodate the adjusted limits.

Substitute the New Limits: Replace the original limits with the new values. Substitute ‘a’ with ‘c’ and ‘b’ with ‘d’ in the integral expression.

Formulas of Limits of Integration

Following are the formulas for the limits of integration:

1. $\int_{a}^{b} f(x) \,dx = \int_{a}^{b} f(t) \,dt$

This says that integrating a function from (a) to (b) with respect to (x) is the same as integrating the same function from (a) to (b) with respect to (t).

2. $\int_{a}^{b} f(x) \,dx = -\int_{b}^{a} f(x) \,dx \int_{a}^{b} f(x) \,dx = -\int_{b}^{a} f(x) \,dx$

This formula means that changing the limits of integration also changes the sign of the integral.

3. $\int_{a}^{b} c \cdot f(x) \,dx = c \cdot \int_{a}^{b} f(x) \,dx$

If you have a constant (c) multiplied by the function, you can take the constant outside the integral.

4. $\int_{a}^{b} [f(x) \pm g(x)] \,dx = \int_{a}^{b} f(x) \,dx \pm \int_{a}^{b} g(x) \,dx$

This formula allows you to split the integral when you have the sum or difference of two functions.

5. $\int_{b}^{a} f(x) \,dx = \int_{a}^{b} f(a + b - x) \,dx$

Integrating from (b) to (a) is the same as integrating from (a) to (b) with a change in the variable.

6. $\int_{a}^{0} f(x) \,dx = \int_{a}^{0} f(a - x) \,dx$

This formula involves integrating from (a) to 0, and it’s related to the one above.

7. $\int_{2a}^{0} f(x) \,dx = 2 \cdot \int_{a}^{0} f(x) \,dx$

If f(2a – x) = f(x), then the integral from (2a) to 0 is twice the integral from (a) to 0.

8. $\int_{2a}^{0} f(x) \,dx = 0$

If f(2a – x) = -f(x), then the integral from (2a) to 0 is zero.

9. $\int_{-a}^{a} f(x) \,dx = 2 \cdot \int_{0}^{a} f(x) \,dx$

If f(x) is an even function f(-x) = f(x), the integral from (-a) to (a) is twice the integral from (0) to (a.

10. $\int_{-a}^{a} f(x) \,dx = 0$

If f(x)is an odd function f(-x) = -f(x), the integral from (-a) to (a) is zero.

Read more about Integral as limit of Sum.

Application of Limits of Integration

Some of the key application of Limits of Integration are:

Physics

Limits of integration help calculate displacement, velocity, and acceleration of objects.
Integration of velocity functions enables the determination of an object’s position or the distance it has traveled.

Economics

Integration plays a crucial role in analyzing production and cost functions.
Limits of integration assist in examining the behavior of variables as quantities approach extremes, aiding in making informed decisions.

Engineering

In fluid mechanics, limits of integration are applied to solve problems related to fluid flow and pressure distribution.
In electrical circuits, limits help analyze signals and determine system behavior.

Read More about Application of Integrals.

Solved Examples of Limits of Integration

Example 1: You have the function f(x) = 2x + 1, and you want to find the area under the curve between the points x = 1 and x = 3. Write and solve the definite integral for this scenario.

Solution:

To solve : I = $\quad \int_{1}^{3} (2x + 1) \,dx$

First, find the antiderivative of (2x+1), which is (x²+x).

⇒I = $\left[x^2 + x\right]_{1}^{3}$

Now substitute the upper limit x=3 and lower limit x=1:

⇒ I = [3² + 3] – [ 1²+ 1]

⇒ I = [9+3] – [1+1]

⇒ I = 12-2

⇒ I = 10

∴ the area under the curve of 2x + 1 from x=1 to x=3 is 10 square units.

Example 2: Consider an object’s velocity given by v(t) = 3t² where (t) is time. Find the distance traveled by the object from t=1 to t=2. Express the solution as a definite integral.

Solution:

Let I = $\quad \int_{1}^{2} 3t^2 \,dt$

First, find the antiderivative of 3t², which is t³.

⇒ I = $\left[t^3\right]_{1}^{2}$

Now substitute the upper limit t=2 and lower limit t=1:

⇒ I = (2)³ – (1)³

⇒ I = 8-1

⇒ I = 7

∴ the distance traveled by the object from (t=1) to (t=2) is 7 units.

Practice Questions on Limits of Integration

Quaetion 1: Evaluate the definite integral $\int_0^2\ \left(4x-1\right)dx$ .

Question 2: Given the function g(t)= 2t³-5t²+3, find the area under the curve from t=1 to t=2. Express the result as a definite integral.

Question 3: A particle’s velocity is represented by v(x)= 3x²-2x+1. Determine the displacement of the particle from x=0 to x=3 using integration.

Limits of Integration: FAQs

1. What is Integration?

In mathematics, Integration refers to finding the integral of a function.

2. What are the Rules of Integration?

Integration rules include the power rule, constant multiple rule, sum/difference rule, substitution rule, and trigonometric rules. These principles guide finding antiderivatives and evaluating definite integrals in calculus.

3. What is the Limit Form of the Integral?

The limit form of the integral, often encountered in calculus, is represented as the limit of a Riemann sum that is given as follows:

$\int_{a}^{b} f(x) \,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x_i$

4. What are the Upper and Lower Limits of Integration?

In definite integration, the upper and lower limits represent the range over which the integration occurs. The lower limit is the starting point, and the upper limit is the endpoint.

5. What are the Integration Formulas for Limits?

Some of the formulas related to limit of integration are:

$\int_{a}^{b} f(x) \,dx = \int_{a}^{b} f(t) \,dt$
$\int_{a}^{b} f(x) \,dx = -\int_{b}^{a} f(x) \,dx \int_{a}^{b} f(x) \,dx = -\int_{b}^{a} f(x) \,dx$
$\int_{a}^{b} c \cdot f(x) \,dx = c \cdot \int_{a}^{b} f(x) \,dx$
$\int_{a}^{b} [f(x) \pm g(x)] \,dx = \int_{a}^{b} f(x) \,dx \pm \int_{a}^{b} g(x) \,dx$

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Methods of Integration

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