Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the definite integral. $$\int_{1}^{3}(2 x+3) d x$$

Step-by-Step Solution

Answer

The definite integral \(\int_{1}^{3}(2x+3) dx\) is equal to 14.

Step 1: Find the antiderivative of the integrand

To find the antiderivative of the function (2x+3), we will use the power rule: the antiderivative of x^n is \(\frac{x^{n+1}}{n+1}\) for any real number n that is not equal to -1. For the term 2x, the current exponent of x is 1. Using the power rule, we get: \[\frac{2x^{1+1}}{1+1} = x^2\] For the constant term 3, we treat this as the antiderivative of \(3x^0\). Using the power rule, we get: \[\frac{3x^{0+1}}{0+1} = 3x\] Now, add the two terms together to find the antiderivative of (2x+3): \[x^2 + 3x\]

Step 2: Evaluate the antiderivative at the upper limit of integration

In this step, we will plug in the upper limit of integration (x=3) into the antiderivative we found in step 1: \[(3^2 + 3(3)) = (9 + 9) = 18\]

Step 3: Evaluate the antiderivative at the lower limit of integration

In this step, we will plug in the lower limit of integration (x=1) into the antiderivative we found in step 1: \[(1^2 + 3(1)) = (1 + 3) = 4\]

Step 4: Subtract the evaluations to find the definite integral

Now, we will subtract the value we got for the antiderivative at the lower limit of integration from the value we got at the upper limit of integration: \[18 - 4 = 14\] The definite integral of \(\int_{1}^{3}(2x+3) dx\) is equal to 14.

Key Concepts

Antiderivative

An antiderivative is a function that reverses the process of differentiation. Basically, if you have a function whose derivative is known, then its antiderivative returns to the original function before it was differentiated.
In the context of definite integrals, the process of finding an antiderivative is the first step.

• The antiderivative provides us with the key to evaluating the definite integral of a function across a specific interval.
• To tackle the problem of integrating \( 2x+3 \), we needed to find the antiderivative of each term in the expression individually.

For example, in our exercise, the antiderivative of the term \( 2x \) uses the basic integration technique, increasing the power of \( x \) by one and dividing by the new power. Meanwhile, the antiderivative of a constant like \( 3 \) is simply the constant times \( x \). Thus, the combined antiderivative is \( x^2 + 3x \). This preparation eventually leads us to solve the definite integral by applying the integration limits.

Power Rule

The power rule is an essential technique in calculus for finding the derivative or antiderivative of functions that consist of power terms. It's a shortcut that simplifies the integration of polynomials.

• The rule states that for any term \( x^n \), the antiderivative is \( \frac{x^{n+1}}{n+1} \), given that \( n eq -1 \).
• This rule is key to solving problems that involve polynomial expressions, such as integrating \( 2x+3 \).

In our specific exercise:
We broke down the task using the power rule:

• For the term \( 2x \), recognized as \( x^1 \), the antiderivative became \( x^2 \) after applying the power rule.
• On the constant \( 3 \), viewing it as \( 3x^0 \) allowed us to integrate it as \( 3x \).

These steps illustrate how powerful and straightforward the power rule is for integrating functions that can be broken down into individual terms.

Integration Limits

Integration limits indicate the boundaries over which we want to evaluate the function. These limits are crucial for computing definite integrals and set them apart from indefinite integrals, which do not have specified limits.

• In our problem, the definite integral is divided by limits from \( x=1 \) to \( x=3 \).
• This interval suggests we compute the net area between the curve \( y = 2x+3 \) and the x-axis between these two points.

The process involves:
1. Solving the antiderivative at the upper limit \( x=3 \), calculated as 18 in the exercise.
2. Evaluating the antiderivative at the lower limit \( x=1 \), which gives 4.
3. Finally, by subtracting the lower limit value from the upper limit value, \( 18 - 4 \), we determine the definite integral to be 14.
Thus, the integration limits provide both the framework and conclusion needed for calculating definite integrals effectively.