Calculator of Definite Integrals

A definite integral calculates the exact area under a curve between two specified points. This calculator computes the definite integral of a function over a given interval using numerical methods.

What is a Definite Integral?

A definite integral represents the signed area between a function's curve and the x-axis over a specified interval [a, b]. It provides exact values for quantities like total distance traveled, accumulated work, or total change in a quantity.

Unlike indefinite integrals, which find antiderivatives, definite integrals give a single numerical result. The definite integral of a function f(x) from a to b is written as ∫[a,b] f(x) dx.

How to Calculate Definite Integrals

Calculating definite integrals involves these steps:

Identify the function f(x) and the interval [a, b]
Find the antiderivative F(x) of f(x)
Evaluate F(x) at the upper limit b and lower limit a
Subtract the lower evaluation from the upper evaluation: ∫[a,b] f(x) dx = F(b) - F(a)

Note

For functions without elementary antiderivatives, numerical methods like Simpson's Rule or the Trapezoidal Rule are used.

The Definite Integral Formula

Definite Integral Formula

∫[a,b] f(x) dx = F(b) - F(a)

Where:

F(x) is the antiderivative of f(x)
a is the lower limit of integration
b is the upper limit of integration

For example, the definite integral of x² from 0 to 2 is calculated as:

Find the antiderivative of x²: (x³)/3
Evaluate at 2: (2³)/3 = 8/3
Evaluate at 0: (0³)/3 = 0
Subtract: 8/3 - 0 = 8/3 ≈ 2.6667

Applications of Definite Integrals

Definite integrals have numerous practical applications including:

Calculating areas under curves
Determining total distance traveled
Finding accumulated work done
Computing total change in a quantity
Calculating average value of a function

Example Applications of Definite Integrals
Application	Example
Area under curve	Calculating the area between a velocity-time graph and the time axis gives total distance traveled
Work done	Integrating force over distance gives the work done by a variable force
Average value	Finding the average temperature over a time period by integrating temperature over time and dividing by the time interval

Frequently Asked Questions

What is the difference between definite and indefinite integrals?: A definite integral calculates a specific area between limits, while an indefinite integral finds a general antiderivative without limits.
When would I use a definite integral calculator?: Use this calculator when you need to compute the exact area under a curve between two points, or when the antiderivative is difficult to find.
Can definite integrals be negative?: Yes, definite integrals can be negative when the function is below the x-axis over the interval, representing a signed area.
What if the function doesn't have an elementary antiderivative?: For such functions, numerical methods like Simpson's Rule or the Trapezoidal Rule can approximate the definite integral.
How accurate are the results from this calculator?: The calculator uses precise numerical methods to provide accurate results for most functions, though some complex functions may require more advanced techniques.