Calculate Integral Online with Limits

This calculator computes definite integrals with specified limits, providing the area under a curve between two points. It's useful for solving problems in physics, engineering, economics, and other fields where accumulation of quantities is important.

What is an Integral with Limits?

An integral with limits, also known as a definite integral, calculates the area under a curve between two specified points (the lower and upper limits). This concept is fundamental in calculus and has applications in various scientific and mathematical disciplines.

The integral of a function f(x) from a to b is written as ∫[a,b] f(x) dx. The result represents the net accumulation of the function's values between the limits.

Integral Formula

∫[a,b] f(x) dx = F(b) - F(a)

Where F(x) is the antiderivative of f(x)

Definite integrals can be computed using geometric methods, numerical approximations, or analytical techniques. This calculator uses numerical integration methods for practical computation.

How to Use This Calculator

Enter the function you want to integrate in the "Function" field. Use standard mathematical notation (e.g., x^2, sin(x), e^x).
Specify the lower limit (a) and upper limit (b) of integration.
Select the number of intervals for numerical computation (higher values give more accurate results but take longer).
Click "Calculate" to compute the integral.
Review the result and visualization of the function and area under the curve.

Note

This calculator uses the trapezoidal rule for numerical integration. For exact results, you may need to compute the antiderivative manually.

The Integral Formula

The definite integral of a function f(x) from a to b is calculated using the antiderivative F(x):

Definite Integral Formula

∫[a,b] f(x) dx = F(b) - F(a)

Where F(x) is the antiderivative of f(x)

The antiderivative F(x) is found by reversing the differentiation process. For example, the antiderivative of x² is (1/3)x³.

Worked Examples

Example 1: Simple Polynomial

Compute ∫[0,2] x² dx

Find the antiderivative: ∫x² dx = (1/3)x³ + C
Evaluate at limits: [(1/3)(2)³] - [(1/3)(0)³] = (8/3) - 0 = 8/3 ≈ 2.6667

Example 2: Trigonometric Function

Compute ∫[0,π] sin(x) dx

Find the antiderivative: ∫sin(x) dx = -cos(x) + C
Evaluate at limits: [-cos(π)] - [-cos(0)] = -(-1) - (-1) = 1 + 1 = 2

Practical Tip

For complex functions, consider using numerical methods when exact antiderivatives are difficult to find.

Frequently Asked Questions

What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration and produces a numerical value representing the area under the curve. An indefinite integral has no limits and produces a family of antiderivatives.

When should I use this calculator?

Use this calculator when you need to compute the area under a curve between two points. It's particularly useful for functions that don't have simple antiderivatives or when you need numerical approximations.

How accurate are the results?

The accuracy depends on the number of intervals used in the numerical integration. More intervals provide better accuracy but require more computation time.