Evaluate The Integral. 3 X2 + 9 Dx 0

Evaluating the Definite Integral: ∫₀³ (3x² + 9) dx

Evaluating definite integrals is a fundamental concept in calculus with wide-ranging applications in various fields like physics, engineering, and economics. This article will provide a comprehensive walkthrough of evaluating the definite integral ∫₀³ (3x² + 9) dx, explaining the steps involved and highlighting key concepts in integral calculus. We'll explore the process from the basics of integration to applying the Fundamental Theorem of Calculus, offering a detailed understanding suitable for both beginners and those looking for a refresher.

Understanding the Problem: ∫₀³ (3x² + 9) dx

The expression ∫₀³ (3x² + 9) dx represents a definite integral. Let's break down its components:

• ∫: This symbol represents the integral sign, indicating the operation of integration.
• (3x² + 9): This is the integrand, the function we are integrating. In this case, it's a polynomial function.
• dx: This indicates that we are integrating with respect to the variable x.
• ₀³: These are the limits of integration. The integral is evaluated from x = 0 to x = 3. This means we are finding the area under the curve of the function (3x² + 9) between these two x-values.

The Power Rule of Integration

Before we tackle the definite integral, we need to understand the power rule of integration, a fundamental technique used to integrate polynomial functions. The power rule states:

∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C

Where:

• 'n' is any real number except -1.
• 'C' is the constant of integration. This is crucial in indefinite integrals (integrals without limits), as it accounts for any constant that could have been added to the original function before differentiation. However, for definite integrals, the constant cancels out.

Step-by-Step Evaluation of the Definite Integral

Now, let's apply the power rule to evaluate our definite integral:

Step 1: Integrate the function:

First, we integrate each term of the integrand separately:

∫(3x² + 9) dx = ∫3x² dx + ∫9 dx

Applying the power rule:

∫3x² dx = 3∫x² dx = 3 * (x³/3) = x³

∫9 dx = 9x

Therefore, the indefinite integral is:

x³ + 9x + C

Step 2: Apply the limits of integration:

This is where we use the Fundamental Theorem of Calculus. The theorem states that if F(x) is an antiderivative of f(x), then:

∫ₐᵇ f(x) dx = F(b) - F(a)

In our case, F(x) = x³ + 9x, a = 0, and b = 3. So, we substitute the upper and lower limits into the antiderivative and subtract:

F(3) = (3)³ + 9(3) = 27 + 27 = 54

F(0) = (0)³ + 9(0) = 0

Therefore:

∫₀³ (3x² + 9) dx = F(3) - F(0) = 54 - 0 = 54

Geometric Interpretation

The result, 54, represents the area under the curve y = 3x² + 9 between x = 0 and x = 3. This area is a combination of a rectangle and a parabolic segment. While calculating the area directly using geometric formulas would be cumbersome here, the integral provides an elegant and accurate solution.

Advanced Concepts and Extensions

While the above method efficiently solves the problem, let's delve into more advanced concepts related to integration:

1. U-Substitution (Integration by Substitution)

U-substitution is a powerful technique for simplifying integrals. It's particularly useful when dealing with composite functions. While not strictly necessary for this specific problem, understanding it broadens your integration toolkit.

For example, if our integrand were more complex, say ∫₀³ 6x(x² + 3)² dx, we could use u-substitution. Let u = x² + 3. Then, du = 2x dx. The integral then simplifies considerably and becomes much easier to evaluate.

2. Integration of Trigonometric Functions

Integrating trigonometric functions requires different techniques. For instance, integrating sin(x) or cos(x) involves fundamental trigonometric identities.

3. Integration by Parts

Integration by parts is crucial when dealing with products of functions. It’s based on the product rule for differentiation and is represented as:

∫u dv = uv - ∫v du

This technique is particularly useful in integrals involving functions like x * sin(x) or x² * eˣ.

4. Partial Fraction Decomposition

For rational functions (fractions of polynomials), partial fraction decomposition breaks down the integrand into simpler fractions, making integration significantly easier.

5. Numerical Integration

For integrals that are difficult or impossible to solve analytically, numerical methods like the Trapezoidal Rule or Simpson's Rule provide approximate solutions. These methods are essential in numerous practical applications where analytical solutions are unavailable.

Applications of Definite Integrals

Definite integrals find applications in various fields:

• Physics: Calculating work done by a force, displacement of an object, and other kinematic quantities.
• Engineering: Determining the area of irregular shapes, calculating volumes of solids of revolution, and solving problems in fluid mechanics.
• Economics: Computing consumer surplus, producer surplus, and other economic indicators.
• Probability and Statistics: Finding probabilities associated with continuous random variables.

Conclusion

Evaluating the definite integral ∫₀³ (3x² + 9) dx is a fundamental exercise in calculus that highlights the power and versatility of integration techniques. By applying the power rule and the Fundamental Theorem of Calculus, we efficiently determined the area under the curve to be 54. Understanding the underlying concepts, as well as the advanced techniques presented, is crucial for tackling more complex integration problems and applying integral calculus to various real-world scenarios. The versatility of integration techniques makes it a cornerstone of mathematical modeling and problem-solving across diverse disciplines. Remember that practice is key to mastering integration; working through various problems and exploring different methods will build your confidence and skill.