107 LearnersLast updated on August 7th, 2025
Math Formula for Definite Integral
In calculus, the definite integral is a fundamental concept that helps to determine the area under a curve between two points. This topic will delve into the formula for calculating definite integrals and its applications.
List of Math Formulas for Definite Integral
The definite integral is a key tool in calculus used to calculate the area under a curve. Let’s learn the formula for calculating definite integrals.
Math Formula for Definite Integral
The definite integral of a function f(x) from a to b is given by the formula: ∫[a, b] f(x) dx = F(b) - F(a) where F is the antiderivative (or indefinite integral) of f.
Properties of Definite Integrals
Definite integrals have several important properties, including:
1. Linearity: ∫[a, b] (cf(x) + g(x)) dx = c∫[a, b] f(x) dx + ∫[a, b] g(x) dx
2. Additivity: ∫[a, c] f(x) dx + ∫[c, b] f(x) dx = ∫[a, b] f(x) dx
3. Reversal of limits: ∫[a, b] f(x) dx = -∫[b, a] f(x) dx
Importance of Definite Integral Formula
In math and real life, we use definite integrals to analyze and understand various phenomena. Here are some important applications of definite integrals:
Definite integrals help calculate areas, volumes, and other quantities in physics and engineering.
By learning this formula, students can easily understand concepts like area under curves, total accumulation, and net change.
Tips and Tricks to Memorize Definite Integral Math Formula
Students often find calculus formulas tricky and confusing. Here are some tips and tricks to master definite integrals:
Understand the geometrical interpretation of definite integrals as area under a curve.
Practice solving integrals to gain familiarity with the process and properties.
Use visualization tools or graphs to see the effect of changing limits and functions.
Real-Life Applications of Definite Integral Math Formula
In real life, definite integrals play a major role in understanding various phenomena. Here are some applications of the definite integral formula:
In physics, to find the displacement of an object, we use definite integrals to calculate the area under a velocity-time graph.
In economics, to determine the consumer surplus, we use definite integrals to calculate the area between demand curve and price level.
Common Mistakes and How to Avoid Them While Using Definite Integral Math Formula
Students make errors when calculating definite integrals. Here are some mistakes and the ways to avoid them, to master them.
Mistake 1
Failing to find the correct antiderivative
Students sometimes struggle to find the correct antiderivative, which leads to errors in calculating the definite integral. Practice finding antiderivatives for different types of functions to improve accuracy.
Mistake 2
Ignoring the limits of integration
When evaluating a definite integral, students may forget to apply the limits of integration. Always remember to substitute the upper and lower limits into the antiderivative to find the integral's value.
Mistake 3
Confusing definite and indefinite integrals
Students often confuse definite integrals with indefinite integrals. Remember that definite integrals have limits and result in a numerical value, while indefinite integrals do not have limits and result in a function plus a constant.
Mistake 4
Incorrectly reversing limits of integration
Reversing the limits of integration changes the sign of the integral. Be cautious when switching the limits to ensure the correct sign of the result.
Mistake 5
Misapplying properties of definite integrals
Students sometimes apply properties, like linearity and additivity, incorrectly. Review the properties of definite integrals and apply them carefully when solving problems.
Examples of Problems Using Definite Integral Math Formula
Problem 1
Calculate the definite integral of f(x) = x^2 from 1 to 3.
The integral is 26/3
Explanation
First, find the antiderivative of f(x) = x^2, which is F(x) = x^3/3.
Then, evaluate F(b) - F(a): F(3) = 3^3/3 = 27/3 = 9 F(1) = 1^3/3 = 1/3 So, ∫[1, 3] x^2 dx = 9 - 1/3 = 26/3
Problem 2
Find the area under the curve f(x) = 2x + 1 from x = 0 to x = 2.
The area is 6
Explanation
First, find the antiderivative of f(x) = 2x + 1, which is F(x) = x^2 + x.
Then, evaluate F(b) - F(a): F(2) = 2^2 + 2 = 6 F(0) = 0^2 + 0 = 0 So, ∫[0, 2] (2x + 1) dx = 6 - 0 = 6
Problem 3
Evaluate the integral of f(x) = 3x from x = 1 to x = 4.
The integral is 27/2
Explanation
First, find the antiderivative of f(x) = 3x, which is F(x) = 3x^2/2.
Then, evaluate F(b) - F(a): F(4) = 3(4^2)/2 = 48/2 = 24 F(1) = 3(1^2)/2 = 3/2 So, ∫[1, 4] 3x dx = 24 - 3/2 = 27/2
Problem 4
Determine the definite integral of f(x) = sin(x) from x = 0 to x = π/2.
The integral is 1
Explanation
First, find the antiderivative of f(x) = sin(x), which is F(x) = -cos(x).
Then, evaluate F(b) - F(a): F(π/2) = -cos(π/2) = 0 F(0) = -cos(0) = -1 So, ∫[0, π/2] sin(x) dx = 0 - (-1) = 1
Problem 5
Calculate the definite integral of f(x) = e^x from x = 0 to x = 1.
The integral is e - 1
Explanation
First, find the antiderivative of f(x) = e^x, which is F(x) = e^x.
Then, evaluate F(b) - F(a): F(1) = e^1 = e F(0) = e^0 = 1 So, ∫[0, 1] e^x dx = e - 1
FAQs on Definite Integral Math Formula
1.What is the formula for a definite integral?
2.What are the properties of definite integrals?
3.How to calculate the area under a curve using definite integrals?
4.What is the difference between definite and indefinite integrals?
5.How do definite integrals apply to real-world problems?
Glossary for Definite Integral Math Formulas
- Definite Integral: A mathematical concept used to calculate the area under a curve between two points.
- Antiderivative: The reverse process of differentiation, also called an indefinite integral.
- Linearity: A property of definite integrals allowing the sum of integrals to be split into separate integrals.
- Additivity: A property that allows combining integrals over adjacent intervals.
- Reversal of Limits: Changing the limits of integration inverts the sign of the integral.
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
