Evaluate The Definite Integral. Calculator

What is a definite integral?

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

Unlike indefinite integrals, which find antiderivatives, definite integrals require both the function and the bounds of integration. The result is a single numerical value representing the accumulation of the function's values over the interval.

How to calculate definite integrals

To evaluate a definite integral:

1. Identify the function f(x) you want to integrate
2. Determine the lower bound a and upper bound b
3. Find the antiderivative F(x) of f(x)
4. Apply the Fundamental Theorem of Calculus: ∫[a,b] f(x) dx = F(b) - F(a)

For complex functions, you may need to use integration techniques like substitution, integration by parts, or partial fractions.

The definite integral formula

This formula represents the difference in the antiderivative evaluated at the upper and lower bounds, giving the exact area under the curve.

Worked examples

Example 1: Simple polynomial

Calculate ∫[1,3] (2x + 1) dx

1. Find the antiderivative: ∫(2x + 1) dx = x² + x + C
2. Evaluate at bounds: (3² + 3) - (1² + 1) = (9 + 3) - (1 + 1) = 11 - 2 = 9
3. Result: The area under the curve is 9 square units

Example 2: Trigonometric function

Calculate ∫[0,π] sin(x) dx

1. Find the antiderivative: ∫sin(x) dx = -cos(x) + C
2. Evaluate at bounds: (-cos(π)) - (-cos(0)) = (1) - (-1) = 2
3. Result: The area under the curve is 2 square units

Interpreting results

The result of a definite integral represents:

• The exact area under the curve between the bounds
• The net accumulation of the function's values over the interval
• The total change in the quantity being measured

When interpreting results, consider the context of your function and the physical meaning of the integral in your specific application.