Problem 6 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\). $$f(x)=4 x-1 ;[2,4]$$

Step-by-Step Solution

Answer

The area of the region under the graph of the function \(f(x) = 4x - 1\) on the interval [2, 4] is 22.

Step 1: Find the integral of the function

To do this, we need to calculate the indefinite integral of the function \(f(x) = 4x - 1\): \[ \int (4x-1) dx \] Now, integrate the function: \[ \int (4x-1) dx = 4\int x dx - \int 1 dx \]

Step 2: Apply the fundamental theorem of calculus

Using the fundamental theorem of calculus, the definite integral of the function \(f(x) = 4x - 1\) on the interval [2, 4] is: \[ \int_{2}^{4} (4x-1) dx = \left[2x^2 - x\right]_{2}^{4} \]

Step 3: Calculate the area using the limits

Now, plug in the values of the limits: Upper Limit(4): \(2(4)^2 - (4) = 32 - 4 = 28\) Lower Limit(2): \(2(2)^2 - (2) = 8 - 2 = 6\) Now subtract the lower limit value from the upper limit value: \[ Area = 28 - 6 = 22 \] So the area of the region under the graph of the function \(f(x) = 4x - 1\) on the interval [2, 4] is 22.

Key Concepts

Definite Integral

The definite integral is a core concept in calculus that helps us calculate the area under a curve within a certain interval. It's denoted as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of the interval, and \( f(x) \) is the function being integrated. The process of finding this area involves

• Evaluating the indefinite integral, which finds the antiderivative of the function.
• Applying the limits of the interval to this antiderivative.

To solve for the area, we substitute these limits into the antiderivative equation:\[ F(b) - F(a) \]Where \( F(x) \) is the antiderivative of \( f(x) \). This subtraction gives us the total area beneath the curve from \( a \) to \( b \), effectively solving the problem of determining the space between the function graph and the x-axis.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a vital bridge between the antiderivative (indefinite integral) and the definite integral. It allows us to calculate the area under a curve, or the definite integral, using the antiderivative. This theorem has two main parts:

• The first part tells us that if a function \( f \) is continuous on an interval \([a, b]\) and \( F \) is its antiderivative, then the definite integral of \( f \) from \( a \) to \( b \) is equal to \( F(b) - F(a) \).
• The second part establishes that the derivative of an integral function \( F(x) \) is the original function \( f(x) \).

Thus, the theorem shows that differentiation and integration are inverse processes. This principle is crucial for effectively calculating areas under curves without needing to resort to ancient geometric approximation methods.

Indefinite Integral

The indefinite integral, unlike the definite integral, does not have specified bounds. It represents a family of functions and is essentially the reverse operation of taking a derivative. When we perform an indefinite integral, we're looking for an antiderivative that, when differentiated, results in the original function. The notation \( \int f(x) \ dx \) suggests finding a function \( F(x) \) such that\[ F'(x) = f(x) \]The result of an indefinite integral includes an added constant \( C \), which accounts for any vertical translations of the function and reflects its general nature. This is because differentiation of a constant \( C \) is zero, so adding any constant doesn't affect the original function's derivative. This integral serves as a stepping stone to the definite integral, providing the base solution needed to apply limits and find specific areas under curves.