Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the graph and find the area of the region bounded by the graph of the function \(f\) and the lines \(y=0, x=a\), and \(x=b\) $$f(x)=4 x^{1 / 3}+x^{4 / 3} ; a=-1, b=8$$

Step-by-Step Solution

Answer

The area of the region bounded by the graph of the function \(f(x)=4x^{\frac{1}{3}}+x^{\frac{4}{3}}\) and the lines \(y=0, x=-1\), and \(x=8\) is \(1941\) square units. To find the area, we used a definite integral: \[A = \int_{-1}^8 \left(4x^{\frac{1}{3}} + x^{\frac{4}{3}}\right) dx\] and applied the power rule for integration, followed by evaluating the integral at the given limits.

Step 1: Sketch the graph

To sketch the graph of the function, input the function \(f(x) = 4x^{\frac{1}{3}} + x^{\frac{4}{3}}\) into a graphing calculator or online graphing tool, such as Desmos or Geogebra. The graph should show the behavior of the function between x = -1 and x = 8.

Step 2: Integrate the function

To find the area of the region bounded by f(x), y=0, x=a, and x=b, we'll need to integrate f(x) between a and b. The area can be found by using the definite integral: \[A = \int_a^b f(x)dx\] In this case, we have: \[A = \int_{-1}^8 \left(4x^{\frac{1}{3}} + x^{\frac{4}{3}}\right) dx\]

Step 3: Apply the power rule to integration

The power rule for integration states that \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) when n ≠ -1. Applying the power rule, we get: \[A = \left[\frac{4x^{\frac{4}{3}}}{\frac{4}{3}} + \frac{x^{\frac{7}{3}}}{\frac{7}{3}}\right]_{-1}^8\]

Step 4: Evaluate the definite integral

Now, we'll need to evaluate the definite integral by substituting the values of a and b, and taking the difference: \[A = \left[\frac{4(8)^{\frac{4}{3}}}{\frac{4}{3}} + \frac{(8)^{\frac{7}{3}}}{\frac{7}{3}}\right] - \left[\frac{4(-1)^{\frac{4}{3}}}{\frac{4}{3}} + \frac{(-1)^{\frac{7}{3}}}{\frac{7}{3}}\right]\] \[A = \left[\frac{144}{\frac{4}{3}} + \frac{4096}{\frac{7}{3}}\right] - \left[\frac{4}{\frac{4}{3}} - \frac{1}{\frac{7}{3}}\right]\] \[A = \left[\frac{216}{1} + \frac{1728}{1}\right] - \left[\frac{6}{1} - \frac{3}{1}\right]\]

Step 5: Calculate the final area

Finally, we'll need to simplify the expression for the total area A: \[A = (216 + 1728) - (6 - 3)\] \[A = 1944 - 3 = 1941\] Therefore, the area of the region bounded by the graph of the function and the lines y=0, x=a, and x=b is 1941 square units.

Key Concepts

Definite Integral

The concept of a definite integral is crucial in calculus. It helps us find the accumulated total, such as distance, area, or volume, over a specific interval. In a nutshell, a definite integral can be seen as the net area under a curve of a function between two points along the x-axis. Suppose we have a continuous function, like \( f(x) \), and two bounds \( a \) and \( b \). The definite integral is expressed as:

• \( \int_a^b f(x) \, dx \)

This notation tells us to sum up the infinite tiny rectangles, the widths of which infinitely approach zero, under the curve of \( f(x) \) from \( x = a \) to \( x = b \). By performing the definite integral for the function \( f(x) = 4x^{\frac{1}{3}} + x^{\frac{4}{3}} \) from \( x = -1 \) to \( x = 8 \), we calculate the area beneath the curve between these bounds.

Area Under a Curve

Calculating the area under a curve is a common application of definite integrals in calculus. This process allows us to find the exact "size" or "space" beneath a function within given limits on the x-axis. Geometrically, this area is under the graph of a continuous function, above the x-axis, and between two vertical lines:

• \( x = a \): starting boundary
• \( x = b \): ending boundary

For the function \( f(x) = 4x^{\frac{1}{3}} + x^{\frac{4}{3}} \), we determined the bounds as \( x = -1 \) and \( x = 8 \). The area derived from integrating this function within these bounds tells us the total space enclosed by the function, its limits, and the x-axis. Here, the area is found by evaluating the definite integral, which calculated as 1941 square units.

Power Rule for Integration

The power rule for integration is a straightforward, essential tool for integrating functions of the form \( x^n \). It states that the integral of \( x^n \) is given by:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

where \( n \) is a real number, and \( C \) represents the constant of integration when calculating indefinite integrals. For definite integrals, like in our exercise, the constant \( C \) is omitted because its effect cancels out.Applying the power rule

• To integrate \( 4x^{1/3} \), we apply: \( \frac{4x^{4/3}}{4/3} \)
• To integrate \( x^{4/3} \), we apply: \( \frac{x^{7/3}}{7/3} \)

These results are then evaluated at the upper and lower limits of the integral to find the desired area under the curve.

Graph of a Function

Sketching the graph of a function gives a visual representation, helping us better understand its behavior and the impact on the calculation of integrals. Features that can be determined from a graph include:

• Intercepts
• Intervals of increase or decrease
• Concavity
• Asymptotic behavior

For the function \( f(x) = 4x^{1/3} + x^{4/3} \), sketching involves using a graphing calculator or tool to visualize it from \( x = -1 \) to \( x = 8 \). By plotting, we see how the function behaves over the given interval, which can reveal patterns in rate changes of the area under the curve. This insight complements the algebraic evaluation when finding the exact area, as we did earlier through the definite integral. Understanding the graph aids us in validating the computed areas.