Calculo Integral Definidas

What is a Definite Integral?

A definite integral represents the signed area between a curve and the x-axis from a lower limit (a) to an upper limit (b). It provides exact values for quantities like total distance traveled, accumulated work, or total change in a function.

Unlike indefinite integrals, which find antiderivatives, definite integrals require both the function and the limits of integration. The result is a single numerical value representing the net area.

Formula

The limits of integration (a and b) must be real numbers with a ≤ b. The function f(x) must be continuous on the closed interval [a, b].

How to Calculate a Definite Integral

1. Identify the function f(x) and the limits of integration (a and b).
2. Find the antiderivative F(x) of f(x).
3. Evaluate F(x) at the upper limit (b) and the lower limit (a).
4. Subtract F(a) from F(b) to get the definite integral value.

Examples

Example 1: Simple Polynomial

Calculate ∫[1,3] (2x + 1) dx.

1. Find the antiderivative: ∫(2x + 1) dx = x² + x + C.
2. Evaluate at limits: (3² + 3) - (1² + 1) = (9 + 3) - (1 + 1) = 11 - 2 = 9.

Example 2: Trigonometric Function

Calculate ∫[0,π] sin(x) dx.

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

Applications

Definite integrals have numerous practical applications in:

• Physics: Calculating work done by a variable force.
• Engineering: Determining the center of mass of a variable-density object.
• Economics: Finding total revenue or cost over a period.
• Probability: Calculating the probability of an event occurring within a range.