Evaluate The Integral. 9 X2 + 81 Dx 0

Evaluating the Definite Integral: ∫₀⁹ (9x² + 81) dx

This article provides a comprehensive guide on how to evaluate the definite integral ∫₀⁹ (9x² + 81) dx. We'll break down the process step-by-step, exploring the fundamental concepts of integration and applying them to solve this specific problem. We will also discuss relevant techniques and considerations for similar integration problems.

Understanding the Problem

The problem asks us to evaluate a definite integral. This means we need to find the area under the curve of the function f(x) = 9x² + 81 between the limits of integration, x = 0 and x = 9. The expression ∫₀⁹ (9x² + 81) dx represents this area calculation.

Key Concepts in Integration

Before diving into the solution, let's review some fundamental concepts:

1. The Indefinite Integral

The indefinite integral, denoted as ∫f(x) dx, represents the family of antiderivatives of the function f(x). An antiderivative, F(x), is a function whose derivative is f(x), i.e., F'(x) = f(x). The indefinite integral always includes an arbitrary constant of integration, typically represented by "+ C".

2. The Power Rule of Integration

The power rule is a crucial tool for integrating polynomial functions. It states that the integral of xⁿ is (xⁿ⁺¹)/(n+1) + C, where n ≠ -1.

3. The Definite Integral

The definite integral, denoted as ∫ₐᵇ f(x) dx, represents the signed area between the curve y = f(x) and the x-axis, from x = a to x = b. It's calculated by finding the antiderivative F(x) and evaluating it at the limits of integration: F(b) - F(a). This is often referred to as the Fundamental Theorem of Calculus.

4. Linearity of Integration

Integration is a linear operation. This means that the integral of a sum is the sum of the integrals, and a constant factor can be pulled out of the integral:

∫[af(x) + bg(x)] dx = a∫f(x) dx + b∫g(x) dx, where 'a' and 'b' are constants.

Step-by-Step Solution: Evaluating ∫₀⁹ (9x² + 81) dx

Now, let's solve the given definite integral:

1. Apply the Linearity of Integration: We can break down the integral into two simpler integrals:

   ∫₀⁹ (9x² + 81) dx = 9∫₀⁹ x² dx + 81∫₀⁹ 1 dx

2. Apply the Power Rule of Integration:

   • For the first integral, ∫₀⁹ x² dx, we apply the power rule with n = 2:

     ∫x² dx = (x³/3) + C

   • For the second integral, ∫₀⁹ 1 dx, we treat 1 as x⁰ and apply the power rule with n = 0:

     ∫1 dx = ∫x⁰ dx = (x¹/1) + C = x + C

3. Evaluate the Indefinite Integrals:

   Substituting the results from step 2 into the equation from step 1:

   9[(x³/3) + C] + 81[x + C] = 3x³ + 9C + 81x + 81C = 3x³ + 81x + 81C + 9C = 3x³ + 81x + 90C

   Notice that the constant of integration (C) appears multiple times. In definite integration, these constants cancel out when we evaluate the antiderivative at the limits.

4. Evaluate the Definite Integral using the Fundamental Theorem of Calculus:

   Now we evaluate the antiderivative, 3x³ + 81x, at the limits of integration (0 and 9):

   [3(9)³ + 81(9)] - [3(0)³ + 81(0)] = [3(729) + 729] - [0] = 2187 + 729 = 2916

Therefore, the value of the definite integral ∫₀⁹ (9x² + 81) dx is 2916.

Further Considerations and Extensions

This problem highlights the fundamental techniques of integration. Let's explore some related concepts and extensions:

1. Visualizing the Area:

The result, 2916, represents the area under the curve of the parabola y = 9x² + 81 between x = 0 and x = 9. You can visualize this area by graphing the function and observing the region.

2. Applications of Definite Integrals:

Definite integrals have numerous applications across various fields, including:

• Physics: Calculating work done by a force, displacement, and other physical quantities.
• Engineering: Determining areas, volumes, and centroids of complex shapes.
• Economics: Computing consumer surplus and producer surplus.
• Probability and Statistics: Finding probabilities and expectations.

3. More Complex Integrals:

While this example involves a relatively simple polynomial function, definite integrals can involve more complex functions, requiring more advanced integration techniques such as substitution, integration by parts, partial fraction decomposition, and trigonometric substitutions.

4. Numerical Integration:

For integrals that are difficult or impossible to solve analytically, numerical methods such as the trapezoidal rule, Simpson's rule, or Gaussian quadrature can be used to approximate the definite integral's value.

5. Error Analysis in Numerical Integration:

When employing numerical integration, understanding the potential sources of error and their magnitude is crucial. Factors such as the step size, the type of numerical method employed, and the function's behavior can all affect the accuracy of the approximation.

6. Software for Integration:

Many software packages, such as Mathematica, Maple, MATLAB, and Wolfram Alpha, have built-in functions for both symbolic and numerical integration, simplifying the process of evaluating complex integrals.

Conclusion

Evaluating the definite integral ∫₀⁹ (9x² + 81) dx involves applying fundamental concepts of integration, primarily the power rule and the linearity of integration. The step-by-step solution demonstrates the process of finding the antiderivative and then evaluating it at the limits of integration to obtain the numerical result of 2916. Understanding these techniques is essential for solving more complex integration problems and appreciating the wide-ranging applications of definite integrals in diverse fields. Remember to always check your work and consider using available tools and resources to ensure accuracy and efficiency in your calculations.