Problem 5 - Essential Calculus

Question

In Exercises \(5-8,\) the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{0}^{4}\left[(x+1)-\frac{x}{2}\right] d x $$

Step-by-Step Solution

Answer

The area between the functions \(x+1\) and \(\frac{x}{2}\) from \(x=0\) to \(x=4\) is 8.

Step 1: Sketch the graphs

First the two functions, \(x+1\) and \(\frac{x}{2}\), need to be plotted on the same graph. The function \(x+1\) is a linear function with a slope of one and a y-intercept of one. The function \(\frac{x}{2}\) is a linear function with a slope of 0.5 and goes through the origin.

Step 2: Shade the region representing the integral

The region represented by the integral is the area between the two functions from \(x=0\) to \(x=4\). So the region between the two functions and between \(x=0\) and \(x=4\) is shaded.

Step 3: Calculate the integral

The integral is calculated as \(\int_{0}^{4}\left[(x+1)-\frac{x}{2}\right] d x\). First break down the integral into two separate integrals \(\int_{0}^{4}(x+1) d x - \int_{0}^{4}\frac{x}{2} d x\). Then calculate each integral individually. The integral of \(x+1\) from \(0\) to \(4\) is \(\frac{1}{2}x^{2}+x\) evaluated from \(0\) to \(4\) which gives 12. The integral of \(\frac{x}{2}\) from \(0\) to \(4\) is \(\frac{1}{2} * \frac{1}{2}x^{2}\) evaluated from \(0\) to \(4\) which gives 4. Subtract these two results to get the value of the original integral which is 12 - 4 = 8.

Step 4: Interpret the result

The value of the integral, which is 8, represents the signed area between the functions \(x+1\) and \(\frac{x}{2}\) over the interval \([0,4]\). In this context, it is the area above the line \(\frac{x}{2}\) and below the line \(x+1\) from \(x=0\) to \(x=4\).

Key Concepts

Area Between Curves

When dealing with integrals that involve a difference of two functions, it's often helpful to visualize the problem by focusing on the area between the curves represented by these functions. In this exercise, we have two linear functions, one is \(x + 1\) and the other is \(\frac{x}{2}\). To find the area between these curves, we first need to sketch them on a graph.
The region between these curves on the interval from \(x = 0\) to \(x = 4\) is what we shade, representing the area of interest in the context of a definite integral. This visual step sets us up to carry out the actual integration by defining clear boundaries for the area we are calculating.

• Sketch both curves on the same set of axes.
• Identify their points of intersection, if any, over the interval in question.
• Shade the region between the curves to clearly see the area represented by the integral.

This approach not only aids in formulating the integral correctly but also enhances intuitive understanding of what the integral represents geometrically.

Linear Functions

Linear functions are among the simplest functions and play a crucial role in calculus, including in problems involving integrals. The functions in our exercise, \(x + 1\) and \(\frac{x}{2}\), are both linear.
A linear function can be generally written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the function \(x + 1\), the slope \(m = 1\) and the y-intercept \(b = 1\), meaning it crosses the y-axis at \(y = 1\).
For \(\frac{x}{2}\), the slope \(m = 0.5\) and the y-intercept \(b = 0\), since it passes through the origin. Understanding the nature and characteristics of these linear functions:

• Big help in predicting their graph's behavior.
• Aids in determining the intersection points.
• Is essential in calculating the area between them correctly.
  

By familiarizing ourselves with the parameters of linear equations, we can proficiently tackle integral problems related to areas between curves.

Integral Calculation

The calculation of the integral \(\int_{0}^{4}[(x+1)-\frac{x}{2}]\,dx\) is a step-by-step process involving several important concepts. First, it's crucial to simplify the integrand by performing the subtraction directly, yielding \(\int_{0}^{4}[x + 1 - \frac{x}{2}]\,dx = \int_{0}^{4}[\frac{2x}{2} + 1 - \frac{x}{2}]\,dx = \int_{0}^{4}[\frac{x}{2} + 1]dx\).
Next, we split this into two easier integrals: \(\int_{0}^{4} (x+1)\,dx\) and \(\int_{0}^{4} \frac{x}{2}\, dx\).

• The integral of \(x+1\) results in \(\frac{1}{2}x^2 + x\).
• The integral of \(\frac{x}{2}\) results in \(\frac{1}{4}x^2\).

Evaluating these integrals from 0 to 4, we find:

• \(\int_{0}^{4}(x+1)\,dx = 12\)
• \(\int_{0}^{4}\frac{x}{2}\,dx = 4\)

Subtract the second result from the first to get the area, which is 12 - 4 = 8.
Thus, the integral tells us the total signed area between the curves across the interval \([0, 4]\).