Evaluating The Integral Of 2x^2 + X - 2 From 1 To 2

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In the realm of calculus, definite integrals play a pivotal role in determining the area under a curve between two specified limits. This article delves into the step-by-step process of evaluating the definite integral of the polynomial function f(x) = 2x² + x - 2 over the interval [1, 2]. We will explore the fundamental concepts, theorems, and techniques involved in solving this problem, providing a clear and concise explanation for students and enthusiasts alike.

Understanding Definite Integrals

Before we embark on the journey of solving the given integral, it is crucial to grasp the essence of definite integrals. In essence, a definite integral represents the net signed area between a curve and the x-axis within a defined interval. The interval is specified by the limits of integration, which in our case are 1 and 2. The definite integral is denoted by the following expression:

∫[a, b] f(x) dx

where:

  • ∫ represents the integral symbol
  • a and b are the lower and upper limits of integration, respectively
  • f(x) is the function being integrated (the integrand)
  • dx indicates that the integration is performed with respect to the variable x

The fundamental theorem of calculus provides the cornerstone for evaluating definite integrals. This theorem establishes a profound connection between differentiation and integration, stating that the definite integral of a function f(x) from a to b can be calculated by finding an antiderivative F(x) of f(x) and evaluating the difference F(b) - F(a). In mathematical terms:

∫[a, b] f(x) dx = F(b) - F(a)

where F'(x) = f(x).

Step-by-Step Solution

Now, let's apply this knowledge to evaluate the definite integral of our given function, f(x) = 2x² + x - 2, from 1 to 2. We will meticulously walk through each step to ensure clarity and understanding.

1. Finding the Antiderivative

The first step in evaluating a definite integral is to determine the antiderivative of the integrand. The antiderivative, also known as the indefinite integral, is a function whose derivative is equal to the original function. In our case, we need to find a function F(x) such that F'(x) = 2x² + x - 2.

To find the antiderivative, we can apply the power rule of integration, which states that the integral of xⁿ is (xⁿ⁺¹)/(n+1), where n is any real number except -1. Applying this rule to each term in our integrand, we get:

∫(2x²) dx = (2x³)/3 + C₁

∫(x) dx = (x²)/2 + C₂

∫(-2) dx = -2x + C₃

where C₁, C₂, and C₃ are constants of integration. Since we are finding the definite integral, the constants of integration will cancel out in the next step, so we can combine them into a single constant C. Thus, the antiderivative of 2x² + x - 2 is:

F(x) = (2x³)/3 + (x²)/2 - 2x + C

2. Evaluating the Antiderivative at the Limits of Integration

The next step involves evaluating the antiderivative F(x) at the upper and lower limits of integration, which are 2 and 1, respectively. This means we need to calculate F(2) and F(1).

Substituting x = 2 into the antiderivative, we get:

F(2) = (2(2)³)/3 + ((2)²)/2 - 2(2) + C

= (16)/3 + 2 - 4 + C

= (16)/3 - 2 + C

Similarly, substituting x = 1 into the antiderivative, we get:

F(1) = (2(1)³)/3 + ((1)²)/2 - 2(1) + C

= (2)/3 + (1)/2 - 2 + C

= (4 + 3 - 12)/6 + C

= -(5)/6 + C

3. Applying the Fundamental Theorem of Calculus

Now, we can apply the fundamental theorem of calculus to find the definite integral. This theorem states that the definite integral of f(x) from 1 to 2 is equal to F(2) - F(1). Therefore:

∫[1, 2] (2x² + x - 2) dx = F(2) - F(1)

= [(16)/3 - 2 + C] - [-(5)/6 + C]

= (16)/3 - 2 + (5)/6

Notice that the constants of integration C cancel out, as we mentioned earlier. Now, we can simplify the expression to find the final value of the definite integral.

= (32 - 12 + 5)/6

= (25)/6

Therefore, the definite integral of 2x² + x - 2 from 1 to 2 is 25/6.

Alternative Approaches and Verification

While we have demonstrated the standard method of evaluating the definite integral, it is worth noting that there are alternative approaches that can be employed. One such method involves using numerical integration techniques, such as the trapezoidal rule or Simpson's rule, to approximate the value of the integral. These methods are particularly useful when the integrand is complex or does not have a readily available antiderivative.

Furthermore, we can verify our result by using computational tools or software that are capable of evaluating definite integrals. These tools provide an independent check on our calculations and can help identify any potential errors.

Importance of Definite Integrals

Definite integrals are not merely abstract mathematical concepts; they have profound applications in various fields of science, engineering, and economics. Some of the key applications include:

  • Calculating areas and volumes: Definite integrals can be used to find the area of regions bounded by curves and the volume of solids of revolution.
  • Determining displacement and distance: In physics, definite integrals can be used to calculate the displacement and distance traveled by an object given its velocity function.
  • Modeling population growth: In biology, definite integrals can be used to model the growth of populations over time.
  • Calculating probabilities: In statistics, definite integrals are used to calculate probabilities associated with continuous random variables.
  • Evaluating financial quantities: In finance, definite integrals can be used to calculate present and future values of investments.

Conclusion

In this article, we have thoroughly explored the process of evaluating the definite integral of the polynomial function f(x) = 2x² + x - 2 over the interval [1, 2]. We have meticulously walked through each step, from finding the antiderivative to applying the fundamental theorem of calculus. We have also discussed alternative approaches and the importance of definite integrals in various fields.

By mastering the concepts and techniques presented in this article, students and enthusiasts can confidently tackle a wide range of definite integral problems. The ability to evaluate definite integrals is a fundamental skill in calculus and is essential for understanding and applying mathematical principles in diverse domains. Remember, practice is key to proficiency, so keep exploring and solving integrals to solidify your understanding.

Further Exploration

To further enhance your understanding of definite integrals, consider exploring the following topics:

  • Applications of definite integrals in various fields
  • Numerical integration techniques (trapezoidal rule, Simpson's rule)
  • Improper integrals (integrals with infinite limits or discontinuous integrands)
  • Multiple integrals (integrals in two or more dimensions)
  • The relationship between definite integrals and differential equations

By delving deeper into these topics, you can gain a more comprehensive understanding of the power and versatility of definite integrals in mathematics and its applications.