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Definite Integrals find the exact area under a curve between specific points like a definite slice of cake. Definite integrals give a single numerical answer. Indefinite integrals ask “What function could have this curve as its slope?”. It represents a whole family of functions and includes an unknown constant like all the cakes that could result in that slice.
Table of Contents
What are Definite Integral?
A definite integral is a mathematical tool used to calculate the exact area under the curve of a function f(x) between two specific limits, a and b. A definite integral can calculate the exact shaded area between the curve, the x-axis, and the two points where x = a and x = b.
Here’s a breakdown of the important symbols.
- ∫: Integral symbol
- a: Lower limit of integration
- b: Upper limit of integration
- f(x): The function we’re integrating
- dx: Infinitesimal change in x (represents the width of tiny slices)
The result of the definite integral is a single numerical value representing the total accumulated area under the curve within those limits.
What is Indefinite Integral?
An Indefinite Integral, unlike its definite counterpart, is all about finding the antiderivative of a function. Imagine a function as the slope of a curve at any given point. An indefinite integral asks, “Which function could have produced this curve as its slope?”
Here’s how it works:
- We denote an indefinite integral with ∫ f(x) dx.
- f(x) is the function we’re integrating (think slope).
- The integral symbol (∫) represents the antiderivative operation.
- dx signifies the variable of integration (often x).
The answer to an indefinite integral is a family of functions, all sharing the property that their derivative is the original function f(x). This family is represented by a general antiderivative F(x) with an added constant to integration, C. This constant accounts for an infinite number of possible functions with the same slope.
In simple terms, a definite integral gives a specific area under a curve, while an indefinite integral gives all the possible functions that could create that curve.
Also Read: Difference Between Parallel and Perpendicular
What is the Difference Between Definite and Indefinite Integrals?
Here is a table summarising the difference between definite and indefinite integrals.
| Feature | Definite Integral | Indefinite Integral |
| Purpose | Calculates the area under a curve between specific limits | Finds the antiderivative of a function |
| Result | Single numerical value (area) | Family of functions (all possible antiderivatives) |
| Notation | ∫_a^b f(x) dx | ∫ f(x) dx |
| Interpretation | The exact area under the curve between x=a and x=b | Represents functions with a given slope |
Difference Between Definite and Indefinite Integrals Formulas
While definite integrals don’t have specific formulas like indefinite integrals, here’s a breakdown of what you need to know.
Indefinite Integral Formulas
These formulas help you find the antiderivative or indefinite integral of a function. It’s helpful to solve indefinite integral problems. Here are some common formulas.
- ∫ 1 dx = x + C (where C is the constant of integration)
- ∫ x^n dx = (x^(n+1))/(n+1) + C (n ≠ -1)
- ∫ e^x dx = e^x + C
- ∫ sin(x) dx = -cos(x) + C
- ∫ cos(x) dx = sin(x) + C
- ∫ sec^2(x) dx = tan(x) + C (and similar formulas for other trigonometric functions)
- ∫ 1/x dx = ln|x| + C
Definite Integral Formulas
The fundamental Theorem of Calculus connects definite and indefinite integrals. Here’s the key formula.
- ∫_a^b f(x) dx = F(b) – F(a)
This formula states that the definite integral of f(x) between a and b is equal to the difference between the values of its antiderivative F(x) evaluated at b and a.
Application of Definite and Indefinite Integrals in Real Life
Definite and indefinite integrals have a wide range of applications in real-life problems. Here’s a breakdown of their real-life uses.
Application of Definite Integrals in Real Life
- Finding Area and Volumes: Calculating areas under curves like land area for irregular plots and volumes of revolution like water displaced by a boat full.
- Finding Work Done: Definite integrals can calculate the work done by a force acting over a distance like the work done by a pump against pressure.
- Finding Average Rates: It helps to determine the average values of a function over an interval like the average speed of a car during a trip.
- Finding Fluid Flow: Definite integrals are used in physics to calculate the total amount of fluid flowing through a pipe over time.
Application of Indefinite Integrals in Real Life
- Finding Motion Analysis: It helps to find displacement (antiderivative of velocity) or total distance travelled (antiderivative of speed).
- Finding Falling Objects: Used to find the distance an object falls under gravity (antiderivative of acceleration due to gravity).
- Finding Electric Current: Indefinite integrals can be used to find the total charge flowing through a circuit (antiderivative of current).
- In Economics: It can model concepts like total cost (antiderivative of marginal cost) or consumer surplus (area between demand curve and price line).
In simple terms, definite integrals give us specific quantities of areas, volumes, and work done, while indefinite integrals help us understand the underlying relationships between changing rates and their total effects.
Also Read: Difference Between Marginal Cost and Average Cost
FAQ’s
A definite integral, which has predetermined limitations to integrate from or to, an indefinite integral does not have any restrictions on integration.
In comparison to Indefinite Integration, Definite Integration is a relatively easier task to do due to its advantageous properties that it has.
It is the total of the integrals that constitutes the integral of a sum. To calculate the integral of the product of a constant and a function, multiply the constant by the integral of the function. This will give you the integral of the product.
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