Problem 73 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Given that \(\int_{-1}^{2} f(x) d x=-2\) and \(\int_{-1}^{2} g(x) d x=3\), evaluate a. \(\int_{-1}^{2}[2 f(x)+g(x)] d x\) b. \(\int_{-1}^{2}[g(x)-f(x)] d x\) c. \(\int_{-1}^{2}[2 f(x)-3 g(x)] d x\)

Step-by-Step Solution

Answer

a. \(-1\) b. \(5\) c. \(-13\)

Step 1: According to the linearity property of integrals, we can split the integral into two parts and factor out the constants: \( \int_{-1}^{2} [2 f(x) + g(x)] dx = 2 \int_{-1}^{2} f(x) dx + \int_{-1}^{2} g(x) dx \) #Step 2: Substituting the given values of integrals#

We know that \(\int_{-1}^{2} f(x) dx = -2\) and \(\int_{-1}^{2} g(x) dx = 3\). Substitute these values into the expression from Step 1: \( 2 (-2) + 3 = -4 + 3 = -1 \) Answer for part a is -1. #b. Evaluating the integral \(\int_{-1}^{2}[g(x)-f(x)] d x\)# #Step 1: Use the linearity property of integrals#

Step 2: According to the linearity property of integrals, we can rewrite the integral as the difference of the original integrals: \( \int_{-1}^{2} [g(x) - f(x)] dx = \int_{-1}^{2} g(x) dx - \int_{-1}^{2} f(x) dx \) #Step 2: Substituting the given values of integrals#

We know that \(\int_{-1}^{2} f(x) dx = -2\) and \(\int_{-1}^{2} g(x) dx = 3\). Substitute these values into the expression from Step 1: \( 3 - (-2) = 3 + 2 = 5 \) Answer for part b is 5. #c. Evaluating the integral \(\int_{-1}^{2}[2 f(x)-3 g(x)] d x\)# #Step 1: Use the linearity property of integrals#

Step 3: According to the linearity property of integrals, we can split the integral into two parts and factor out the constants: \( \int_{-1}^{2} [2 f(x) - 3 g(x)] dx = 2 \int_{-1}^{2} f(x) dx - 3 \int_{-1}^{2} g(x) dx \) #Step 2: Substituting the given values of integrals#

We know that \(\int_{-1}^{2} f(x) dx = -2\) and \(\int_{-1}^{2} g(x) dx = 3\). Substitute these values into the expression from Step 1: \( 2(-2) - 3(3) = -4 - 9 = -13 \) Answer for part c is -13.

Key Concepts

Linearity of Integrals

In integral calculus, one of the key properties of integrals is their linearity. This property makes it easier to solve complex integrals by breaking them down into simpler parts. Linearity means that the integral of a sum of functions is equal to the sum of the integrals of the functions. Specifically, if we have two functions, \(f(x)\) and \(g(x)\), and constants \(a\) and \(b\), the linearity property tells us:

• \(\int [a \cdot f(x) + b \cdot g(x)] \, dx = a \int f(x) \, dx + b \int g(x) \, dx\).

This property allows constant factors to be moved outside the integral sign, which simplifies calculations. By applying linearity, we can evaluate integrals like \(\int_{-1}^{2}[2 f(x) + g(x)] \, dx\) by calculating \(2 \int_{-1}^{2} f(x) \, dx + \int_{-1}^{2} g(x) \, dx\). It's like breaking down a complicated task into smaller, manageable steps.

Definite Integrals

Definite integrals provide the accumulation of quantities, and they are represented with specific upper and lower limits of integration. For instance, \(\int_{-1}^{2} f(x) \, dx\) computes the net area under the curve \(f(x)\) from \(x = -1\) to \(x = 2\). Distinguishing between definite and indefinite integrals is important. While indefinite integrals lack limits and include a constant of integration, definite integrals have limits and result in a numerical value.

• Definite integrals calculate the "net" sum of areas, accounting for parts of the curve above and below the x-axis.
• The resulting value can be positive, negative, or zero, depending on the function \(f(x)\).

In our examples, definite integrals were used to solve for results like \(-2\), \(3\), and in combined forms like \(\int_{-1}^{2}[2 f(x) + g(x)] \, dx\). These integrations were evaluated numerically, providing useful insight into the behavior of the functions over the defined interval.

Mathematical Problem Solving

Mathematical problem solving with integrals often involves a structured approach. By applying mathematical properties like linearity and using definite integrals, problems can be methodically tackled.

• First, identify and understand the problem. In this case, we are given integrals of two functions, \(f(x)\) and \(g(x)\), to solve composite integrals.
• Next, use the property of linearity to simplify your integral expressions before substituting values.
• Finally, perform calculations by substituting known integral values and solve the equation step-by-step to reach the solution.

Let's look at part \((a)\) of our exercise: starting with \(\int_{-1}^{2}[2 f(x)+g(x)] \, dx\), we applied the linearity rule to break it into simpler parts. Then, using the values provided, \(\int_{-1}^{2} f(x) \, dx = -2\) and \(\int_{-1}^{2} g(x) \, dx = 3\), we calculated each part separately and combined them to find the answer, \(-1\). This methodical problem-solving approach is essential for tackling calculus problems effectively.