Problem 45 - Calculus for Biology and Medicine

Question

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{-1}^{2}(2-|x|) d x $$

Step-by-Step Solution

Answer

The value of the integral is 1.5.

Step 1: Understand the Absolute Value Function

The integral \( \int_{-1}^{2}(2-|x|) \, dx \) involves the absolute value function \(|x|\). This function creates a "V" shape centered at the origin. For \(x \geq 0\), \(|x| = x\). For \(x < 0\), \(|x| = -x\).

Step 2: Break Down the Function

Rewrite the function \(2 - |x|\) based on the behavior of the absolute value. For \(x < 0\), \(|x| = -x\), hence \(2 - |x| = 2 + x\). For \(x \geq 0\), \(|x| = x\), hence \(2 - |x| = 2 - x\).

Step 3: Set Up the Piecewise Function

The function \(2 - |x|\) can be expressed as a piecewise function: \(2 - |x| = \begin{cases} 2 + x, & \text{if } x < 0 \ 2 - x, & \text{if } x \geq 0 \end{cases} \)

Step 4: Calculate the Area Under Each Piece

Calculate the area for each segment: - For \(x \in [-1, 0]\), we have a right triangle with base 1 and height 1 (slope goes from \(y = 1\) to \(y = 2\)). The area is \(0.5 \times 1 \times 1 = 0.5\).- For \(x \in [0, 2]\), we have a larger triangle with base 2 and height 2 (slope goes from \(y = 2\) to \(y = 0\)). The area is \(0.5 \times 2 \times 2 = 2\).

Step 5: Applying the Sign Rule

When considering the integral as a signed area, note that for the region \(x < 0\), the x-axis is above the function, which gives us a negative area. Hence, the area should be taken as \(-0.5\) for the region \([-1,0]\). Add both areas considering the sign:\(-0.5 + 2 = 1.5\).

Key Concepts

Understanding the Area Under a Curve

To grasp the concept of the integral as the area under a curve, it's important to recognize that integrals let us find the area between the curve and the x-axis within given bounds. For positive values of a function above the x-axis, the area under the curve is calculated normally as a positive value. This simply means we treat the problem like finding the area of a shape on a piece of paper. For regions where the curve is below the x-axis, or for negative values of the integrand when considered in isolation, the area is considered negative. This is akin to cutting out a section from our paper representation.
In this exercise, the function is split into two key areas between the limits (-1, 2) based on the piecewise nature of the function derived from the absolute value function.

• From -1 to 0, the curve is below the x-axis. Thus, we find the area of this section as a triangle and apply a negative sign.
• From 0 to 2, the curve is above the x-axis, making this section positive.

Finally, these areas are summed, considering their signs to find the total integral result.

The Absolute Value Function Simplified

The absolute value function \(|x|\) can seem tricky but is actually pretty straightforward once understood. It is defined as the non-negative value of x, meaning it measures how far a number is from zero on a number line. It does not matter whether x is positive or negative, |x| will always be positive since distance can't be negative. This function presents a "V" shape on a graph. When dealing with integrals involving \(|x|\), we often need to break down the function to understand how it behaves on different intervals.
In the problem at hand, because \(|x|\) changes behavior at x = 0, the function \(2 - |x|\) needs to be split:

• For x values less than 0, \(|x|\) becomes -x. So, \(2 - |x|\) translates to \(2 + x\).
• For x values greater or equal to 0, \(|x|\) remains x, which gives us \(2 - x\).

This step is essential before setting up an integral to properly assess the areas involved.

Exploring Piecewise Functions

Piecewise functions are functions that define different expressions based on distinct intervals of the input variable. They are used especially in cases where the function behaves differently over different ranges. In our problem, the function \(2 - |x|\) is expressed as a piecewise function:
\[ 2 - |x| = \begin{cases} 2 + x, & \text{if } x < 0 \2 - x, & \text{if } x \geq 0 \end{cases} \]
This means we use one formula for x less than zero and another for x equal to or greater than zero.Understanding how to set up and integrate piecewise functions allows for accurate calculations of areas or other properties over different segments. Piecewise form helps to simplify integration by breaking it into manageable intervals where the function behaves consistently. You apply the area formulas separately for each interval segment and sum them up, taking care to factor in each segment's sign based on its position relative to the x-axis.