Problem 25 - Calculus Single Variable

Question

Evaluate the given definite integral by finding an antiderivative of the integrand and applying Theorem \(3 .\) $$ \int_{1}^{2}\left(6 x^{2}-2 x\right) d x $$

Step-by-Step Solution

Answer

The value of the definite integral is 11.

Step 1: Identify the Integrand

The integral we need to evaluate is \( \int_{1}^{2}(6x^2 - 2x) \, dx \). The integrand here is the function \( 6x^2 - 2x \).

Step 2: Find the Antiderivative

To integrate \( 6x^2 - 2x \), we find the antiderivative by integrating each term separately. The antiderivative of \( 6x^2 \) is \( 2x^3 \), and the antiderivative of \( -2x \) is \( -x^2 \). Thus, the antiderivative of \( 6x^2 - 2x \) is \( 2x^3 - x^2 \).

Step 3: Apply the Limits of Integration

Substitute the upper and lower limits of integration into the antiderivative. This results in evaluating \( \left[ 2x^3 - x^2 \right]_{1}^{2} \).

Step 4: Compute the Definite Integral

Evaluate the expression at the upper limit, \( x = 2 \), giving \( 2(2)^3 - (2)^2 = 16 - 4 = 12 \). Then, evaluate it at the lower limit, \( x = 1 \), giving \( 2(1)^3 - (1)^2 = 2 - 1 = 1 \). The definite integral is \( 12 - 1 = 11 \).

Key Concepts

Antiderivative

In calculus, the antiderivative is a concept that refers to a function whose derivative is the given function. Essentially, finding an antiderivative reverses the process of differentiation. When dealing with definite integrals, such as the one in our exercise, you'll first need to find the antiderivative of the integrand. This is a crucial step to solve integrals.Here's the basic idea:

• When you differentiate a function and then find its antiderivative, you end up with the original function, plus a constant.
• For example, if the derivative of a function is given by \( f'(x) = 6x^2 - 2x \), then one of the antiderivatives is \( F(x) = 2x^3 - x^2 \), as seen in our exercise.

Remember, the antiderivative represents a family of functions. In definite integrals, the constant of integration cancels out when evaluating the limits. Hence, while a constant is usually added, it is excluded when solving definite integrals.

Limits of Integration

The limits of integration are the values that define the interval over which you are integrating. In our exercise, these limits are \( x = 1 \) and \( x = 2 \). These limits are crucial because they determine the specific part of the graph under the curve you are evaluating.When applying the antiderivative to these limits:

• First, substitute the upper limit (in our case, \( x = 2 \)) into the antiderivative \( 2x^3 - x^2 \).
• Next, substitute the lower limit (here, \( x = 1 \)) into the antiderivative.
• Finally, subtract the value obtained from the lower limit from the value obtained from the upper limit.

This calculation gives you the definite integral, which represents the net area under the curve between the specified limits. Using limits of integration helps turn the general antiderivative into a specific numeric value.

Integrand

The term 'integrand' refers to the function that is to be integrated. It is the focus of the integration process and determining it correctly is crucial for finding the antiderivative. In our exercise, the integrand is the function \( 6x^2 - 2x \).Understanding the integrand involves:

• Recognizing the algebraic expression within the integral sign.
• Determining its structure and breaking it down into manageable parts for integration.
• Applying the integration process term by term, if necessary, as seen in our example.

Correctly identifying the integrand ensures that the rest of the integration process follows smoothly, allowing us to apply limits of integration effectively and derive results correspondingly.

Calculus Problem Solving

Solving calculus problems, like evaluating definite integrals, involves a structured approach combining several concepts. To work through the exercise efficiently, follow these key steps:

Firstly, identify the integrand, as this sets the stage for finding its antiderivative.

Next, find the antiderivative by integrating term by term. Ensure each derivative is reversed correctly to its original form.

Apply the limits of integration once the antiderivative is determined to find the specific numeric result intended by a definite integral.

Finally, compute the definite integral by substituting and evaluating the values at the upper and lower limits, ensuring subtraction is correctly calculated to achieve the final answer.

Mastering calculus problem solving involves consistent practice of these steps and comprehension of each concept in isolation and together.