Integral 2xdx From 10 To 13

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

This article provides a detailed explanation of how to solve the definite integral ∫₁₀¹³ 2x dx, covering the fundamental concepts, step-by-step calculations, and practical applications. We'll explore the underlying principles of integration, delve into the power rule of integration, and demonstrate how to evaluate definite integrals using the Fundamental Theorem of Calculus. Furthermore, we'll discuss the geometric interpretation of the integral and explore some real-world applications of this type of calculation.

Understanding the Definite Integral

Before diving into the solution, let's establish a firm understanding of what a definite integral represents. A definite integral is a mathematical tool used to calculate the signed area between a curve and the x-axis over a specified interval. In our case, the curve is represented by the function f(x) = 2x, and the interval is from x = 10 to x = 13. The "signed area" means that areas below the x-axis are considered negative, while areas above the x-axis are positive.

Key Concepts:

• Integrand: The function being integrated, in this case, 2x.
• Limits of Integration: The values that define the interval of integration, 10 and 13 in this example. These represent the lower and upper bounds of the integral.
• Differential (dx): Indicates the variable with respect to which we are integrating.

Applying the Power Rule of Integration

The power rule is a fundamental technique for integrating polynomial functions. The rule states:

∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C

where 'n' is any real number except -1, and 'C' is the constant of integration. In our case, we have ∫2x dx, which can be rewritten as 2∫x¹ dx.

Applying the power rule:

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

Notice that we've added the constant of integration, 'C'. This is crucial for indefinite integrals (integrals without limits). However, for definite integrals, the constant of integration cancels out during the evaluation process.

Evaluating the Definite Integral

Now we'll evaluate the definite integral using the Fundamental Theorem of Calculus. The theorem states that if F(x) is an antiderivative of f(x), then:

∫_a^b f(x) dx = F(b) - F(a)

In our case, f(x) = 2x, and its antiderivative is F(x) = x² (we've omitted the constant of integration 'C' as mentioned earlier). The limits of integration are a = 10 and b = 13. Therefore:

∫₁₀¹³ 2x dx = F(13) - F(10) = (13)² - (10)² = 169 - 100 = 69

Therefore, the definite integral of 2x dx from 10 to 13 is 69.

Geometric Interpretation

The definite integral ∫₁₀¹³ 2x dx represents the area of a trapezoid. The function 2x is a straight line with a positive slope. The trapezoid is bounded by the x-axis, the line y = 2x, and the vertical lines x = 10 and x = 13. The area of a trapezoid is given by:

Area = (1/2) * (base1 + base2) * height

In our case, base1 = 2(10) = 20, base2 = 2(13) = 26, and height = 13 - 10 = 3. Therefore:

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

This confirms our previous calculation using the Fundamental Theorem of Calculus. The geometric interpretation provides a visual understanding of what the definite integral represents.

Real-World Applications

Definite integrals, like the one we've solved, have numerous real-world applications across various fields:

1. Physics:

• Calculating displacement: If 2x represents the velocity of an object at time x, then the definite integral represents the displacement (change in position) of the object between time 10 and time 13.
• Determining work done: If 2x represents a force acting on an object over a distance x, then the integral represents the total work done.

2. Engineering:

• Calculating areas: Definite integrals are extensively used to find areas of irregular shapes, which is essential in civil and mechanical engineering.
• Determining volumes: They're also used in calculating volumes of solids of revolution.

3. Economics:

• Calculating consumer surplus: In economics, definite integrals help to calculate the consumer surplus, which measures the difference between the maximum price a consumer is willing to pay and the actual price they pay.
• Calculating producer surplus: Similarly, it calculates producer surplus, which measures the difference between the actual price and the minimum price a producer is willing to accept.

4. Statistics and Probability:

• Calculating probabilities: Definite integrals are crucial in probability theory for finding probabilities associated with continuous random variables.
• Calculating expected values: They're also used in determining expected values for continuous probability distributions.

Conclusion

We've comprehensively explored the solution to the definite integral ∫₁₀¹³ 2x dx, emphasizing the underlying principles and practical applications. Through the step-by-step calculation, the geometric interpretation, and the discussion of real-world examples, we've demonstrated the power and versatility of definite integrals in solving various mathematical and scientific problems. Understanding definite integrals is a foundational concept in calculus and has broad implications across numerous fields. This in-depth analysis aims to equip readers with a thorough understanding of this important mathematical tool. Remember that mastering the fundamental concepts, like the power rule and the Fundamental Theorem of Calculus, is key to successfully tackling more complex integration problems.