Evaluating The Definite Integral Of (15x^2 - 2x + 7) From 3 To 9

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Introduction

In this article, we will delve into the process of evaluating definite integrals, a fundamental concept in calculus. Specifically, we will tackle the definite integral of the polynomial function f(x) = 15x² - 2x + 7 over the interval [3, 9]. This exercise will not only demonstrate the application of the power rule for integration but also highlight the significance of the Fundamental Theorem of Calculus in determining the precise value of the integral. Understanding definite integrals is crucial for various applications in physics, engineering, economics, and other scientific fields, where calculating areas under curves, accumulated changes, and average values are essential.

The ability to solve definite integrals is a cornerstone of calculus, enabling us to move beyond mere algebraic manipulations to explore the dynamic relationships described by functions. Mastering this skill opens doors to understanding more complex mathematical models and real-world phenomena. The steps involved in evaluating a definite integral, such as finding the antiderivative and applying the limits of integration, offer valuable insights into the nature of functions and their behavior. This article aims to provide a comprehensive guide to these steps, ensuring a clear understanding of the underlying principles and techniques.

By the end of this discussion, you will not only be able to evaluate this specific integral but also gain a deeper appreciation for the power and versatility of definite integrals in mathematical analysis. So, let's embark on this mathematical journey and unravel the intricacies of integral calculus.

Understanding Definite Integrals

To truly grasp the process of evaluating definite integrals, it's essential to first understand what they represent and how they differ from indefinite integrals. A definite integral, denoted as ∫ab f(x) dx, essentially calculates the net signed area between the curve of the function f(x) and the x-axis, within the interval defined by the limits of integration, 'a' and 'b'. In simpler terms, it quantifies the accumulated effect of the function f(x) over the specified interval.

In contrast, an indefinite integral, represented as ∫ f(x) dx, yields a family of functions known as antiderivatives. These antiderivatives differ only by a constant, often denoted as 'C'. While the indefinite integral provides a general representation of the function's integral, the definite integral gives a specific numerical value. This value is obtained by evaluating the antiderivative at the upper and lower limits of integration and then subtracting the results.

The Fundamental Theorem of Calculus forms the bedrock of evaluating definite integrals. It elegantly connects differentiation and integration, asserting that the definite integral of a function can be computed by finding its antiderivative and evaluating it at the limits of integration. This theorem provides a powerful and efficient method for calculating areas, volumes, and other quantities that would otherwise be cumbersome to determine.

The significance of definite integrals extends far beyond theoretical mathematics. They are indispensable tools in various fields, including:

  • Physics: Calculating displacement from velocity, work done by a force, and the center of mass of an object.
  • Engineering: Determining the stress and strain in structures, analyzing fluid flow, and designing electrical circuits.
  • Economics: Modeling economic growth, calculating consumer surplus, and analyzing financial markets.
  • Statistics: Finding probabilities and expected values in probability distributions.

By understanding the concept of definite integrals and their applications, we can appreciate their role in solving real-world problems and gaining insights into complex systems. The next section will focus on the specific steps involved in evaluating the definite integral given in the problem statement.

Steps to Evaluate the Definite Integral

Now, let's embark on the journey of evaluating the definite integral ∫39 (15x² - 2x + 7) dx. This process involves a series of well-defined steps that leverage the power of integral calculus. We'll break down each step to ensure clarity and comprehension.

Step 1: Find the Antiderivative

The first and foremost step is to determine the antiderivative of the integrand, which is the function being integrated (15x² - 2x + 7 in our case). The antiderivative, denoted as F(x), is a function whose derivative is equal to the integrand. To find the antiderivative, we employ the power rule of integration, which states that ∫xn dx = (xn+1) / (n+1) + C, where n is any real number except -1, and C is the constant of integration.

Applying the power rule to each term of the integrand, we get:

  • ∫15x² dx = 15 * (x3 / 3) = 5x³
  • ∫-2x dx = -2 * (x² / 2) = -x²
  • ∫7 dx = 7x

Combining these results, the antiderivative of (15x² - 2x + 7) is F(x) = 5x³ - x² + 7x + C. Note that we include the constant of integration, C, as there are infinitely many antiderivatives that differ only by a constant. However, as we'll see in the next step, this constant will cancel out when evaluating the definite integral.

Step 2: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus provides the bridge between antiderivatives and definite integrals. It states that if F(x) is an antiderivative of f(x), then ∫ab f(x) dx = F(b) - F(a). In other words, the definite integral of f(x) from a to b is equal to the difference between the antiderivative evaluated at b and the antiderivative evaluated at a.

In our case, a = 3 and b = 9. Therefore, we need to calculate F(9) and F(3) using the antiderivative we found in Step 1, F(x) = 5x³ - x² + 7x + C.

  • F(9) = 5(9)³ - (9)² + 7(9) + C = 5(729) - 81 + 63 + C = 3645 - 81 + 63 + C = 3627 + C
  • F(3) = 5(3)³ - (3)² + 7(3) + C = 5(27) - 9 + 21 + C = 135 - 9 + 21 + C = 147 + C

Now, we subtract F(3) from F(9) to find the value of the definite integral:

∫39 (15x² - 2x + 7) dx = F(9) - F(3) = (3627 + C) - (147 + C) = 3627 - 147 = 3480

Notice that the constant of integration, C, cancels out during the subtraction. This is a crucial observation, as it implies that the value of the definite integral is independent of the choice of the antiderivative. We could have chosen any value for C, and the result would have been the same.

Step 3: Interpret the Result

The final step is to interpret the result in the context of the problem. The value of the definite integral, 3480, represents the net signed area between the curve of the function f(x) = 15x² - 2x + 7 and the x-axis, within the interval [3, 9]. Since the result is positive, it indicates that the area above the x-axis is greater than the area below the x-axis within this interval.

In summary, we have successfully evaluated the definite integral by finding the antiderivative, applying the Fundamental Theorem of Calculus, and interpreting the result. This process demonstrates the power and elegance of integral calculus in quantifying areas and accumulated changes.

Detailed Calculation of the Antiderivative

As we've established, finding the antiderivative is a pivotal step in evaluating definite integrals. Let's delve deeper into the process of calculating the antiderivative of our integrand, f(x) = 15x² - 2x + 7. This involves applying the power rule of integration and understanding how it works for polynomial terms.

The power rule of integration states that the integral of xn with respect to x is given by (xn+1) / (n+1) + C, where n is any real number except -1, and C is the constant of integration. This rule is a direct consequence of the power rule of differentiation, which states that the derivative of xn is nxn-1. Integration, being the reverse process of differentiation, essentially