Actividad 2 Calculo Integral Tecmilenio

Introduction

Integral calculus is a fundamental branch of mathematics that deals with the concept of integration, which is the reverse process of differentiation. It has numerous applications in physics, engineering, economics, and other sciences. This guide focuses on the key concepts and techniques covered in Actividad 2 Cálculo Integral from TecMilenio.

The activity typically includes:

• Basic integral rules and formulas
• Techniques of integration (substitution, integration by parts, partial fractions)
• Applications of integrals (area under curves, volume of solids, work done by a variable force)
• Practice exercises and solutions

Basic Integrals

Before diving into complex techniques, it's essential to master the basic rules of integration. Here are some fundamental integral formulas:

These basic integrals form the foundation for more advanced techniques. Practice applying these rules to simple functions to build confidence before moving on to more complex problems.

Techniques of Integration

When basic integrals don't suffice, more advanced techniques are needed. Here are three common methods:

Integration by Substitution

This technique, also known as u-substitution, is useful when the integrand is a composite function. The general approach is:

1. Identify an inner function u and its derivative du.
2. Rewrite the integral in terms of u.
3. Integrate with respect to u.
4. Substitute back in terms of the original variable.

Integration by Parts

This method is based on the product rule for differentiation and is particularly useful for integrals of products of polynomials and transcendental functions. The formula is:

The choice of u and dv is crucial. A common strategy is to use the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to select u.

Partial Fractions

This technique is used to integrate rational functions by expressing them as sums of simpler fractions. The general approach is:

1. Factor the denominator into linear and irreducible quadratic factors.
2. Express the integrand as a sum of partial fractions.
3. Integrate each partial fraction separately.

Applications

Integrals have numerous practical applications in various fields. Here are three common uses:

Area Under Curves

The definite integral of a function over an interval [a, b] gives the area between the curve and the x-axis. For functions that are negative or change sign, the total area is the sum of the absolute values of the integrals over the respective intervals.

Volume of Solids

The volume of a solid of revolution can be found using the disk or shell method. The disk method is used when the function is rotated about a horizontal axis, while the shell method is used for rotation about a vertical axis.

Work Done by a Variable Force

When a variable force F(x) moves an object along the x-axis from x = a to x = b, the work done is given by the integral of F(x) from a to b.

Practice Exercises

To reinforce your understanding, try solving these practice problems. Solutions are provided in the next section.

1. ∫(3x² + 2x - 1) dx
2. ∫x eˣ dx
3. ∫sin(2x) dx
4. ∫(x + 1)/(x² + x) dx
5. Find the area under the curve y = x² from x = 0 to x = 2.

Solutions

Here are the solutions to the practice exercises:

1. ∫(3x² + 2x - 1) dx = x³ + x² - x + C
2. ∫x eˣ dx = (x - 1)eˣ + C
3. ∫sin(2x) dx = (-1/2)cos(2x) + C
4. ∫(x + 1)/(x² + x) dx = ln|x| + C
5. The area under y = x² from 0 to 2 is 8/3.