Teorema Fundamental Del Calculo Integral

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus, developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, bridges the gap between differentiation and integration. It states two main principles:

1. The first part states that differentiation and integration are inverse operations.
2. The second part provides a method to evaluate definite integrals using antiderivatives.

This theorem is foundational in calculus and has numerous applications in mathematics, physics, engineering, and other sciences.

First Part of the Theorem

The first part of the Fundamental Theorem of Calculus states that if a function \( f \) is continuous on the closed interval \([a, b]\), and \( F \) is the antiderivative of \( f \) on \([a, b]\), then:

This means that the antiderivative \( F \) of \( f \) is the same as the integral of \( f \) from \( a \) to \( x \). The derivative of \( F \) is \( f \):

This establishes that differentiation and integration are inverse operations.

Second Part of the Theorem

The second part of the Fundamental Theorem of Calculus provides a method to evaluate definite integrals. If \( f \) is continuous on \([a, b]\) and \( F \) is an antiderivative of \( f \) on \([a, b]\), then:

This means that the definite integral of \( f \) from \( a \) to \( b \) can be found by evaluating the antiderivative \( F \) at the endpoints \( b \) and \( a \), and subtracting these values.

Applications of the Theorem

The Fundamental Theorem of Calculus has numerous practical applications:

• Calculating areas under curves
• Determining volumes of solids of revolution
• Finding average values of functions
• Solving differential equations
• Analyzing motion and velocity

These applications make the theorem essential in various fields of science and engineering.

Worked Example

Let's evaluate the integral \( \int_{1}^{3} 2x \, dx \) using the Fundamental Theorem of Calculus.

1. Find the antiderivative \( F(x) \) of \( f(x) = 2x \).
2. \( F(x) = x^2 + C \) (where \( C \) is the constant of integration).
3. Evaluate \( F \) at the upper and lower limits: \( F(3) = 3^2 = 9 \) and \( F(1) = 1^2 = 1 \).
4. Subtract the lower limit evaluation from the upper limit evaluation: \( 9 - 1 = 8 \).

The value of the integral is 8.