Evaluate The Integral. 9 X2 + 81 Dx 0

Evaluating the Definite Integral: ∫(9x² + 81)dx from 0 to 9

This article provides a comprehensive guide on how to evaluate the definite integral ∫(9x² + 81)dx from 0 to 9. We'll break down the process step-by-step, explaining the underlying concepts and techniques involved in integral calculus. This will cover not only the solution but also contextualize it within the broader framework of integration, making it useful for students and anyone looking to refresh their understanding of this fundamental mathematical operation.

Understanding Definite Integrals

Before diving into the solution, let's establish a clear understanding of what a definite integral represents. A definite integral, denoted as ∫_a^b f(x) dx, calculates the signed area between the curve of a function f(x) and the x-axis, bounded by the limits of integration 'a' and 'b'. In simpler terms, it sums up infinitely small slices of area under the curve. The 'a' and 'b' represent the lower and upper limits of integration, respectively. The result of a definite integral is a number, unlike indefinite integrals which result in a function.

Key Concepts:

• Integrand: The function being integrated (in this case, 9x² + 81).
• Limits of Integration: The values that define the interval over which the integration is performed (0 and 9 in this example).
• Antiderivative: The reverse of a derivative; finding an antiderivative is the core process of integration.

Step-by-Step Solution: ∫(9x² + 81)dx from 0 to 9

Let's now systematically solve the definite integral: ∫₀⁹ (9x² + 81) dx.

Step 1: Find the Antiderivative

The first step involves finding the antiderivative of the integrand, 9x² + 81. We do this by applying the power rule of integration:

∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where 'n' is any real number except -1, and 'C' is the constant of integration.

Applying this rule to each term of the integrand:

• ∫9x² dx: Here, n = 2. So, the antiderivative is (9x²⁺¹)/(2+1) = 3x³
• ∫81 dx: Here, we can consider this as 81x⁰ dx. Applying the power rule, we get 81x.

Therefore, the antiderivative of 9x² + 81 is 3x³ + 81x. We don't need to include the constant of integration 'C' when dealing with definite integrals because it cancels out during the evaluation process.

Step 2: Evaluate the Antiderivative at the Limits of Integration

Now, we substitute the upper and lower limits of integration (9 and 0, respectively) into the antiderivative we found in Step 1:

• Upper Limit (x = 9): 3(9)³ + 81(9) = 2187 + 729 = 2916
• Lower Limit (x = 0): 3(0)³ + 81(0) = 0

Step 3: Subtract the Lower Limit Value from the Upper Limit Value

The final step involves subtracting the value of the antiderivative at the lower limit from its value at the upper limit:

2916 - 0 = 2916

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

Visualizing the Integral: Geometric Interpretation

The result, 2916, represents the total area under the curve y = 9x² + 81 between x = 0 and x = 9. This area is composed of a parabola (9x²) and a rectangle (81). While we've calculated the area using integration, it's insightful to visualize this geometrically. The area is a combination of a parabolic segment and a rectangle, reinforcing the meaning behind the numerical result.

Applications of Definite Integrals

Definite integrals have numerous applications across various fields. Here are a few examples:

• Calculating Areas: As demonstrated in this problem, definite integrals provide a powerful tool for calculating areas of complex shapes bounded by curves and lines.

• Finding Volumes of Solids of Revolution: By rotating a curve around an axis, we can calculate the volume of the resulting solid using integration techniques, extending the concept beyond simple two-dimensional areas.

• Physics and Engineering: Definite integrals are crucial in physics and engineering for calculating quantities like work done by a force, displacement, and center of mass.

• Probability and Statistics: Definite integrals are fundamental to probability theory and statistics, helping to compute probabilities associated with continuous random variables.

• Economics: They play a significant role in economics, for example, calculating consumer surplus or producer surplus in market analysis.

Further Exploration: Techniques of Integration

While this example used the basic power rule of integration, more complex integrals might require other techniques:

• Substitution: Used to simplify integrals by substituting a new variable.

• Integration by Parts: A technique used for integrating products of functions.

• Partial Fraction Decomposition: Used for integrating rational functions (fractions of polynomials).

• Trigonometric Substitutions: Employed when dealing with integrals involving trigonometric functions.

Mastering these techniques broadens the scope of integrals that can be evaluated.

Conclusion: The Power of Definite Integration

This article detailed the evaluation of the definite integral ∫₀⁹ (9x² + 81) dx, demonstrating the step-by-step process and explaining the underlying mathematical concepts. The solution highlights the power of definite integrals in calculating areas and solving problems across diverse disciplines. Understanding the fundamental principles of integration and mastering various integration techniques are essential for anyone working with calculus and its applications in science, engineering, and other quantitative fields. The ability to visualize the integral geometrically also provides valuable insights into the meaning and interpretation of the results, making this a crucial concept for a comprehensive understanding of calculus. This skill, combined with a grasp of the various integration techniques, will provide a strong foundation for tackling even more complex problems in the future.