Online Integration Calculator with Limits

What is Integration with Limits?

Integration is a fundamental concept in calculus that represents the accumulation of quantities. When we talk about integration with limits, we're referring to definite integrals, which calculate the exact area under a curve between two specified points.

Definite integrals are written as:

Integration with limits is used in various fields including physics, engineering, economics, and statistics to calculate areas, volumes, work done, and more.

How to Use This Calculator

Using our online integration calculator with limits is simple:

1. Enter the integrand function in the first field (e.g., "x^2" for x²)
2. Specify the lower limit (a) in the second field
3. Specify the upper limit (b) in the third field
4. Click "Calculate" to get the result

The calculator will display the definite integral result and show a graphical representation of the function and the area under the curve between the specified limits.

The Integration Formula

The fundamental theorem of calculus connects differentiation and integration. For a continuous function f(x) with an antiderivative F(x), the definite integral from a to b is given by:

This formula tells us that the definite integral of a function between two points is equal to the difference in the antiderivative evaluated at those points.

Worked Examples

Example 1: Simple Polynomial

Calculate ∫[1 to 3] x² dx

1. Find the antiderivative of x²: (x³)/3 + C
2. Evaluate at upper limit (3): (3³)/3 = 9
3. Evaluate at lower limit (1): (1³)/3 = 1/3
4. Subtract: 9 - (1/3) = 26/3 ≈ 8.6667

Example 2: Trigonometric Function

Calculate ∫[0 to π] sin(x) dx

1. Find the antiderivative of sin(x): -cos(x) + C
2. Evaluate at upper limit (π): -cos(π) = -(-1) = 1
3. Evaluate at lower limit (0): -cos(0) = -1
4. Subtract: 1 - (-1) = 2