Integral 2xdx From 10 To 13

Definite Integral of 2x dx from 10 to 13: A Comprehensive Guide

The seemingly simple definite integral ∫₁₀¹³ 2x dx might appear straightforward at first glance. However, a thorough understanding involves not just calculating the numerical result but also grasping the underlying concepts and exploring various applications. This comprehensive guide will delve into the intricacies of this integral, covering its calculation, graphical representation, and practical implications.

Understanding the Definite Integral

Before tackling the specific problem, let's establish a solid foundation in the principles of definite integration. The definite integral ∫_a^b f(x) dx represents the signed area between the curve of the function f(x), the x-axis, and the vertical lines x = a and x = b. The "signed" aspect means areas above the x-axis are considered positive, while areas below are negative.

Key Concepts:

• Antiderivative: The process of definite integration involves finding the antiderivative (or indefinite integral) of the function. The antiderivative, F(x), is a function whose derivative is f(x).
• Fundamental Theorem of Calculus: This theorem provides the link between differentiation and integration. It states that the definite integral of a function f(x) from a to b is equal to the difference in the values of its antiderivative F(x) evaluated at b and a: ∫_a^b f(x) dx = F(b) - F(a).

Calculating ∫₁₀¹³ 2x dx

Now, let's apply these concepts to our specific problem: ∫₁₀¹³ 2x dx.

1. Find the Antiderivative:

The antiderivative of 2x is found by applying the power rule of integration: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration. In our case, n = 1:

∫2x dx = 2 * ∫x dx = 2 * (x²/2) + C = x² + C

2. Apply the Fundamental Theorem of Calculus:

Using the Fundamental Theorem of Calculus, we evaluate the antiderivative at the upper and lower limits of integration:

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

Therefore, the definite integral ∫₁₀¹³ 2x dx equals 69. This represents the area under the curve y = 2x between x = 10 and x = 13.

Graphical Representation

Visualizing the integral can provide a deeper understanding. The function y = 2x is a straight line with a slope of 2 passing through the origin. The definite integral represents the area of a trapezoid bounded by the line y = 2x, the x-axis, and the vertical lines x = 10 and x = 13.

You can easily sketch this trapezoid. The trapezoid's vertices will be at the points (10, 20), (13, 26), (13, 0), and (10, 0). The area calculation confirms the result we obtained through integration:

Area of Trapezoid = (1/2) * (sum of parallel sides) * height = (1/2) * (20 + 26) * 3 = 69

Applications and Interpretations

The definite integral ∫₁₀¹³ 2x dx, and the resultant value of 69, can have various interpretations depending on the context.

1. Area Calculation: As demonstrated earlier, it directly represents the area under the curve y = 2x between x = 10 and x = 13.

2. Accumulation of a Quantity: If 2x represents a rate of change of a quantity (e.g., the rate of water flowing into a tank in liters per hour), then the integral represents the total accumulated quantity between the time intervals t = 10 hours and t = 13 hours. In this scenario, 69 would represent the total volume of water accumulated in the tank during those 3 hours.

3. Work Done: In physics, if 2x represents a force function, the integral would represent the work done by the force over a given distance.

4. Average Value: The average value of the function 2x over the interval [10, 13] can also be calculated using the definite integral:

Average Value = (1/(b-a)) * ∫_a^b f(x) dx = (1/(13-10)) * 69 = 23

This means the average value of the function 2x between x = 10 and x = 13 is 23.

Extending the Understanding: Related Concepts

Let's explore some related concepts that build upon the understanding of ∫₁₀¹³ 2x dx.

1. Indefinite Integrals: While this problem focused on the definite integral, it's crucial to understand the concept of indefinite integrals. The indefinite integral ∫2x dx = x² + C represents a family of functions, each differing by a constant C. The constant of integration is crucial when dealing with indefinite integrals but is eliminated when evaluating definite integrals.

2. Numerical Integration: For more complex functions where finding the antiderivative is difficult or impossible, numerical methods such as the Trapezoidal Rule, Simpson's Rule, or Monte Carlo integration can be employed to approximate the definite integral. These methods are particularly valuable when dealing with experimental data or functions that lack analytical antiderivatives.

3. Applications in different fields: The applications of definite integrals extend to a wide range of fields including physics (calculating work, displacement, and other quantities), engineering (determining areas, volumes, and centroids), economics (computing total revenue or cost), and probability (calculating probabilities and expected values). The fundamental principle remains the same: calculating the area under a curve, which holds a significant meaning depending on the context.

Conclusion: Beyond the Calculation

While the numerical result of ∫₁₀¹³ 2x dx = 69 is important, a complete understanding involves grasping the underlying theoretical concepts, visualizing the integral graphically, and exploring the various practical interpretations and applications of the result. This deep understanding is crucial for tackling more complex integration problems and applying the concept in diverse fields of study and real-world scenarios. The ability to interpret the outcome of an integral in relation to the given problem is vital, surpassing mere calculation and touching upon the core meaning of the mathematical operation. This approach allows for a more comprehensive and insightful comprehension of calculus and its practical applications.