Problem 33 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the integral by interpreting it in terms of areas. $$\int_{-1}^{2}|x| d x$$

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{5}{2}\).

Step 1: Understand the Absolute Value Function

The function \(|x|\) represents the absolute value of \(x\), which means that it is equal to \(x\) when \(x \geq 0\) and \(-x\) when \(x < 0\).

Step 2: Determine the Interval Segments

The integral is from \(-1\) to \(2\). The absolute value function changes at \(x = 0\). Thus, we should break the integral into two parts: from \(-1\) to \(0\) and from \(0\) to \(2\).

Step 3: Evaluate the Integral from -1 to 0

From \(-1\) to \(0\), the absolute value \(|x|\) is equal to \(-x\). We find the area under the curve \(-x\) from \(-1\) to \(0\): \[\int_{-1}^{0} -x \, dx = \left[ -\frac{x^2}{2} \right]_{-1}^{0} = 0 - \left( -\frac{1}{2} \right) = \frac{1}{2}.\]

Step 4: Evaluate the Integral from 0 to 2

From \(0\) to \(2\), the absolute value \(|x|\) is simply \(x\). Calculate the area under the curve \(x\) from \(0\) to \(2\):\[\int_{0}^{2} x \, dx = \left[ \frac{x^2}{2} \right]_{0}^{2} = \frac{4}{2} - 0 = 2.\]

Step 5: Add the Areas Together

To evaluate the complete integral from \(-1\) to \(2\), add the areas calculated in Steps 3 and 4:\[\frac{1}{2} + 2 = \frac{5}{2}.\]

Key Concepts

Absolute Value Function

The absolute value function, denoted as \(|x|\), is an essential concept used widely in calculus and mathematics as a whole. Understanding this function is crucial when dealing with calculations involving distance or measuring deviation from zero.
Absolute value describes the magnitude of a number without considering its sign. For any real number \(x\), the absolute value \(|x|\) is defined as:

• \(x\), if \(x \geq 0\)
• \(-x\), if \(x < 0\)

Essentially, \(|x|\) is the distance of the number \(x\) from zero on the number line. It makes every negative number positive, thereby focusing purely on magnitude.
For example, if \(x = -3\), then \(|x| = -(-3) = 3\). When solving integrals involving absolute value functions, it's necessary to determine where the function changes its form. For instance, in the function \(f(x) = |x|\), this change occurs at \(x = 0\). Recognizing such points helps in appropriately breaking the integral into manageable parts.

Area Under Curves

Calculus provides tools to find the area under curves of functions, which is a fundamental idea used to determine many real-life quantities, such as distance, probability, and more.
The area under a curve between two points \(a\) and \(b\) is often found using definite integrals. When interpreting definite integrals in terms of area, the result represents the "net area" between the curve and the x-axis over a specified interval.
The notion here is to consider positive areas above the x-axis as positive and those below it as negative:

• When the curve is above the x-axis, it contributes positively to the overall area.
• When the curve is below the x-axis, it contributes negatively.

The purpose is to calculate the exact accumulated area, considering both stretches of the function. For absolute value functions, understanding the area under each segment of the function is paramount, as shown in the provided example with \(\int_{-1}^{2}|x| \; dx\). The curve needs segmentation into parts where the function has distinct behaviors, like above and below the x-axis.

Definite Integrals

Definite integrals are integral calculus expressions that calculate the accumulation of values such as areas, volumes, and other concepts between two points on a curve. In mathematical notation, a definite integral is expressed as \(\int_{a}^{b} f(x) \, dx\), which evaluates the net area under a curve \(f(x)\) from \(x = a\) to \(x = b\).
A definite integral uses the following components:

• The integrand \(f(x)\), which is the function being integrated.
• The limits of integration \(a\) and \(b\), which define the interval of interest.
• The differential \(dx\), indicating the variable of integration.

One must first handle the absolute value within the integrand by analyzing how the function is structured within the provided limits. For an integral like \(\int_{-1}^{2}|x| dx\), it's vital to split the integration into two segments \([-1, 0]\) and \([0, 2]\), because the absolute value function behaves differently over these intervals.
After breaking down the integral, each part is evaluated separately, and then, the parts are summed to find the total area. Each sub-integral provides a piece of the total area under the specific segments of the function. By aggregating these areas, the solution to the definite integral is achieved, offering insight into the scope of the problem.