Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-22, calculate \(\int_{a}^{b} f(x) d x\), where a and \(b\) are the left and right end points for which \(f\) is defined, by using the Interval Additive Property and the appropriate area formulas from plane geometry. Begin by graphing the given function. \(f(x)= \begin{cases}3 x & \text { if } 0 \leq x \leq 1 \\ 2(x-1)+2 & \text { if } 1

Step-by-Step Solution

Answer

The integral is 2.5.

Step 1: Graph the Function

Let's start by graphing the piecewise function. The first segment, \( f(x) = 3x \), is a straight line from \( x = 0 \) to \( x = 1 \). The second segment, \( f(x) = 2(x-1) + 2 = 2x \), is a line from \( x = 1 \) to \( x = 2 \). Plot these lines over their respective intervals on the coordinate plane.

Step 2: Calculate the Area of the First Segment

The first segment is defined by \( f(x) = 3x \) over the interval \( [0, 1] \). The graph forms a right triangle with the x-axis as one leg, the line \( x = 1 \) as the other leg, and the line connecting \( (0,0) \) to \( (1,3) \). The area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). Here, base = 1 and height = 3, so the area is \( \frac{1}{2} \times 1 \times 3 = 1.5 \).

Step 3: Calculate the Area of the Second Segment

The second segment is \( f(x) = 2x \) over \( (1, 2] \). This portion of the graph forms another right triangle above the line from \( (1,2) \) to \( (2,4) \). With base = 1 and height = 2, the area is \( \frac{1}{2} \times 1 \times 2 = 1 \).

Step 4: Apply Interval Additive Property

The Interval Additive Property allows us to add the areas from both segments since the function is piecewise continuous. Thus, the total area under the function from \( x = 0 \) to \( x = 2 \) is the sum of the individual areas calculated: \( 1.5 + 1 = 2.5 \).

Key Concepts

Piecewise Functions

Piecewise functions are a type of mathematical function where different expressions are applied to different intervals of the domain. In this exercise, the function is defined using two separate rules:

• For the interval \(0 \leq x \leq 1\), the function is given by \(f(x) = 3x\).
• For the interval \(1 < x \leq 2\), the rule changes to \(f(x) = 2x\).

This type of function is essential in modeling real-world scenarios where a single rule is not sufficient. Understanding piecewise functions involves recognizing when each rule applies and how to deal with points of transition between different rules.

Area Under a Curve

Calculating the area under a curve is a fundamental concept in integral calculus. For piecewise functions, this involves treating each segment as a separate entity. In our problem, we have two distinct segments:

• The first segment between \(x = 0\) and \(x = 1\) forms a triangular area.
• The second segment from \(x = 1\) to \(x = 2\) forms another triangular area.

To find the total area under the piecewise function, each section is calculated independently, using basic geometry for simple shapes like triangles, and then added together. This allows you to understand the contribution of each piece of the function to the total area.

Interval Additive Property

The Interval Additive Property is a crucial principle when working with definite integrals and piecewise functions. It dictates that the integral of a function over an interval can be broken down into sub-intervals and computed separately. Specifically, for our exercise, you can calculate:

• The integral from \(x = 0\) to \(x = 1\)
• The integral from \(x = 1\) to \(x = 2\)

Once computed, you simply add these numbers together to get the integral over the full interval from \(x = 0\) to \(x = 2\). This property is what allows us to handle piecewise functions so effectively in calculus.

Graphing Functions

Graphing functions is an important step to visually understand the problem and confirm calculations. For piecewise functions, each segment is plotted according to its rule in its specified interval. In our case:

• First, graph \(f(x) = 3x\) from \(x = 0\) to \(x = 1\), a line segment ascending at a steady rate.
• Next, plot \(f(x) = 2x\) from \(x = 1\) to \(x = 2\), giving another line segment.

Plotting these line segments clearly shows the exact areas under consideration, helping to visualize the concept of calculating the area under piecewise functions and providing a reference for validating integral calculations.