Integrate 2x Dx From 10 To 13

Integrating 2x dx from 10 to 13: A Comprehensive Guide

This article provides a detailed explanation of how to integrate the function 2x with respect to x, evaluating the definite integral from the lower limit of 10 to the upper limit of 13. We will cover the fundamental concepts of integration, the power rule of integration, and the process of evaluating definite integrals. We will also explore the application of this type of integral in various fields.

Understanding the Fundamentals of Integration

Integration is a fundamental concept in calculus, essentially the reverse process of differentiation. While differentiation finds the instantaneous rate of change of a function, integration finds the area under a curve. This area represents the accumulation of the function over a specified interval. There are two main types of integrals:

• Indefinite Integrals: These represent the family of functions whose derivative is the integrand. They always include a constant of integration, denoted by "+C," because the derivative of a constant is zero.

• Definite Integrals: These represent the area under a curve between two specified limits (the lower and upper limits of integration). The constant of integration is not needed in definite integrals because it cancels out during the evaluation process.

The Power Rule of Integration

The power rule is a crucial tool for integrating polynomial functions. The rule states:

∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1

This rule essentially increases the exponent of the variable by one and then divides by the new exponent. The "+C" represents the constant of integration for indefinite integrals.

Solving the Definite Integral: ∫₁₀¹³ 2x dx

Our task is to evaluate the definite integral ∫₁₀¹³ 2x dx. Let's break down the process step-by-step:

Step 1: Apply the Power Rule

First, we apply the power rule of integration to the integrand, 2x. We can rewrite 2x as 2x¹. Applying the power rule, we get:

∫2x dx = 2 * ∫x¹ dx = 2 * (x¹⁺¹)/(1+1) = 2 * (x²)/2 = x²

Step 2: Evaluate the Indefinite Integral

At this stage, we have the indefinite integral: x² + C. Note that the constant of integration "+C" is included because it's an indefinite integral.

Step 3: Evaluate the Definite Integral

Now, we evaluate the definite integral using the fundamental theorem of calculus. This theorem states that the definite integral of a function f(x) from a to b is given by F(b) - F(a), where F(x) is the antiderivative of f(x). In our case, f(x) = 2x and F(x) = x². Therefore, we have:

∫₁₀¹³ 2x dx = [x²]₁₀¹³ = (13²) - (10²) = 169 - 100 = 69

Therefore, the value of the definite integral ∫₁₀¹³ 2x dx is 69.

Graphical Interpretation

The definite integral ∫₁₀¹³ 2x dx represents the area under the curve y = 2x between x = 10 and x = 13. This area forms a trapezoid. The area of a trapezoid is given by:

Area = (1/2) * (sum of parallel sides) * height

In this case, the parallel sides are the y-values at x = 10 and x = 13, which are 20 and 26, respectively. The height is the difference between the x-values, which is 13 - 10 = 3. Therefore:

Area = (1/2) * (20 + 26) * 3 = (1/2) * 46 * 3 = 69

This confirms our earlier result obtained through integration.

Applications of Definite Integrals

Definite integrals have numerous applications across various fields, including:

1. Physics

• Calculating displacement from velocity: If you have a function representing the velocity of an object over time, the definite integral of that function gives the displacement of the object over a specific time interval.

• Calculating work done by a force: The work done by a variable force is given by the definite integral of the force function over the displacement.

• Calculating the center of mass: Definite integrals are crucial in determining the center of mass of a body or a system of particles.

2. Engineering

• Calculating areas of irregular shapes: Definite integrals provide a powerful method to find areas of irregular shapes that cannot be easily calculated using standard geometric formulas. This is essential in civil and mechanical engineering.

• Calculating volumes of solids of revolution: The volumes of objects generated by revolving a curve around an axis can be calculated using definite integrals. This is important in design and manufacturing.

• Determining stress and strain: Definite integrals are used in the calculation of stress and strain distributions in structures, which is critical in structural analysis.

3. Economics

• Calculating consumer surplus and producer surplus: These economic concepts, which represent the net benefit to consumers and producers, are calculated using definite integrals.

• Calculating total revenue and total cost: Definite integrals can be used to calculate total revenue and total cost functions from marginal revenue and marginal cost functions.

4. Probability and Statistics

• Calculating probabilities: In probability theory, definite integrals are used to calculate the probabilities associated with continuous random variables.

• Calculating expected values: Definite integrals play a critical role in calculating expected values for continuous random variables.

Further Exploration of Integration Techniques

While the power rule is sufficient for simple polynomial functions like 2x, other functions require more advanced integration techniques. These include:

• Integration by substitution (u-substitution): This technique simplifies integrals by substituting a new variable, making them easier to solve.

• Integration by parts: This technique is used to integrate products of functions, often involving exponential or trigonometric functions.

• Partial fraction decomposition: This method is used to integrate rational functions (fractions of polynomials).

• Trigonometric substitutions: These substitutions are employed when integrating functions involving square roots of quadratic expressions.

Conclusion

This comprehensive guide explains how to integrate the function 2x from 10 to 13, illustrating the fundamental principles of integration and showcasing the practical applications of definite integrals across diverse fields. Mastering integration techniques is crucial for anyone pursuing studies or careers in mathematics, science, engineering, economics, or related disciplines. The ability to solve definite integrals enables the precise calculation of areas, volumes, and other quantities of interest. Remember that while the example presented here is relatively straightforward, more complex functions will require the application of more advanced integration techniques, underscoring the importance of continued learning and practice in this fundamental area of calculus.