Evaluate ∫ (3x^2 - 2) Dx: A Step-by-Step Guide
In this comprehensive guide, we will walk you through the process of evaluating the definite integral ∫ₐ¹ (3x² - 2) dx. This type of problem is a staple in calculus, and mastering it will significantly enhance your understanding of integral calculus. We'll break down each step, ensuring you grasp the underlying concepts and can confidently tackle similar problems. Our goal is to provide a clear, concise, and easy-to-follow explanation, avoiding the use of decimals in our final answer. Let’s dive in!
Understanding Definite Integrals
Before we jump into solving the specific integral, let's briefly review what a definite integral represents. In simple terms, a definite integral calculates the signed area between a curve and the x-axis over a specified interval. The integral ∫ₐ¹ (3x² - 2) dx, for instance, asks us to find the area under the curve y = 3x² - 2 from x = a to x = 1. The values 'a' and '1' are called the limits of integration. Understanding this geometric interpretation can make the process more intuitive.
When approaching any definite integral problem, it’s vital to remember the Fundamental Theorem of Calculus. This theorem connects differentiation and integration, allowing us to evaluate definite integrals using antiderivatives. We’ll use this theorem extensively throughout our solution. The process involves finding the antiderivative of the function inside the integral, and then evaluating this antiderivative at the upper and lower limits of integration. The difference between these values gives us the value of the definite integral. So, with this understanding, we are well-prepared to solve the integral at hand.
Remember, the key to mastering calculus is practice. So, as we walk through this example, try to understand the logic behind each step. Don’t just memorize the process; focus on comprehending why each step is necessary and how it contributes to the final solution. The more you understand, the better equipped you'll be to tackle more complex problems in the future. We’re here to guide you through each step, making the process as clear and intuitive as possible. Let's move forward and start the actual evaluation of the integral.
Step 1: Finding the Antiderivative
The first critical step in evaluating the definite integral is to find the antiderivative of the function 3x² - 2. An antiderivative, also known as the indefinite integral, is a function whose derivative is equal to the original function. In other words, we are looking for a function F(x) such that F'(x) = 3x² - 2. To find the antiderivative, we'll use the power rule for integration and the constant rule.
The power rule for integration states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1 and C is the constant of integration. Applying this rule to 3x², we get:
∫3x² dx = 3 * ∫x² dx = 3 * (x²⁺¹)/(2+1) + C₁ = 3 * (x³)/3 + C₁ = x³ + C₁
Next, we need to find the antiderivative of the constant term -2. The rule for integrating a constant is ∫k dx = kx + C, where k is a constant. Therefore:
∫-2 dx = -2x + C₂
Now, we combine these two antiderivatives to find the antiderivative of the entire function 3x² - 2:
∫(3x² - 2) dx = ∫3x² dx - ∫2 dx = x³ - 2x + C
Here, C represents the constant of integration, which is the sum of C₁ and C₂. While it's important to remember the constant of integration for indefinite integrals, it will cancel out when we evaluate the definite integral, so we can often omit it for the sake of brevity in the following steps. This step of finding the antiderivative is crucial because it sets the stage for applying the Fundamental Theorem of Calculus. Make sure you understand the power rule and constant rule for integration, as they are fundamental tools in calculus. With the antiderivative in hand, we are ready to move on to the next step: evaluating the antiderivative at the limits of integration.
Step 2: Applying the Fundamental Theorem of Calculus
Now that we have found the antiderivative, x³ - 2x, we can apply the Fundamental Theorem of Calculus to evaluate the definite integral. The Fundamental Theorem of Calculus states that if F(x) is an antiderivative of f(x), then:
∫ₐ¹ f(x) dx = F(1) - F(a)
In our case, f(x) = 3x² - 2 and F(x) = x³ - 2x. The limits of integration are a and 1. So, we need to evaluate F(1) and F(a) and then subtract F(a) from F(1).
First, let's evaluate F(1):
F(1) = (1)³ - 2(1) = 1 - 2 = -1
Next, we evaluate F(a):
F(a) = (a)³ - 2(a) = a³ - 2a
Now, according to the Fundamental Theorem of Calculus, we subtract F(a) from F(1):
∫ₐ¹ (3x² - 2) dx = F(1) - F(a) = -1 - (a³ - 2a)
This step is the heart of evaluating definite integrals. The Fundamental Theorem of Calculus provides a direct link between antiderivatives and the value of the definite integral. It transforms the problem from finding an area under a curve to a straightforward algebraic calculation. By plugging the limits of integration into the antiderivative and finding the difference, we effectively compute the signed area. Make sure you understand how this theorem works and its significance in calculus. We are now in the final stage of the solution, where we simplify the expression to obtain the final answer.
Step 3: Simplifying the Expression
Having applied the Fundamental Theorem of Calculus, we have the expression -1 - (a³ - 2a). The final step is to simplify this expression to get our answer in the most concise form. This involves distributing the negative sign and combining like terms.
Let’s start by distributing the negative sign in front of the parentheses:
-1 - (a³ - 2a) = -1 - a³ + 2a
Now, we can rearrange the terms to present the expression in a more standard format, typically with the highest power of the variable first:
-1 - a³ + 2a = -a³ + 2a - 1
This simplified expression, -a³ + 2a - 1, represents the value of the definite integral ∫ₐ¹ (3x² - 2) dx. Notice that the answer is in terms of 'a', as the lower limit of integration is a variable. This means the value of the integral will change depending on the value of 'a'. If we were given a specific value for 'a', we could substitute it into this expression to get a numerical result.
Simplifying the expression is an essential part of the process. It ensures that your answer is clear, easy to understand, and ready for any further calculations or analysis. This algebraic manipulation showcases the importance of having a strong foundation in algebra when working with calculus problems. With this simplified expression, we have successfully evaluated the definite integral without using decimals, fulfilling the requirements of the problem. We have reached the conclusion of our comprehensive guide.
Conclusion
In conclusion, we have successfully evaluated the definite integral ∫ₐ¹ (3x² - 2) dx, arriving at the simplified expression -a³ + 2a - 1. We accomplished this by following a step-by-step approach:
- Finding the antiderivative of the function 3x² - 2, which is x³ - 2x.
- Applying the Fundamental Theorem of Calculus, which involves evaluating the antiderivative at the upper and lower limits of integration and finding the difference.
- Simplifying the resulting expression to obtain the final answer.
Each step is crucial, and understanding the logic behind each one is key to mastering integral calculus. Remember, the Fundamental Theorem of Calculus is a cornerstone of calculus, linking differentiation and integration in a profound way. Practice applying this theorem to various definite integrals to build your proficiency.
This process not only answers the specific question but also reinforces your understanding of definite integrals and the techniques involved in their evaluation. Calculus is a subject that builds upon itself, so each problem you solve strengthens your foundation for future challenges. By working through this example, you’ve gained valuable experience and insights into the world of integral calculus. Keep practicing, keep exploring, and you’ll continue to grow your mathematical skills.
For further exploration and practice with calculus concepts, consider visiting Khan Academy's Calculus section. This resource offers a wealth of information, exercises, and videos to help you deepen your understanding of calculus.
