Problem 32 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the integral by interpreting it in terms of areas. $$\int_{-1}^{3}(3-2 x) d x$$

Step-by-Step Solution

Answer

The integral evaluates to 4.

Step 1: Identify the function

The given function to integrate is \(f(x) = 3 - 2x\). This is a linear function that represents a straight line on the graph. To interpret it in terms of areas, plot the line \(y = 3 - 2x\) between \(x = -1\) and \(x = 3\).

Step 2: Determine the x-intercept

Find the x-intercept of the function by setting \(f(x) = 0\):\[3 - 2x = 0\]\[2x = 3\]\[x = \frac{3}{2}\]This shows that the x-intercept is at \((\frac{3}{2}, 0)\).

Step 3: Identify areas bounded by the x-axis

The line \(3 - 2x\) crosses the x-axis at \(x = \frac{3}{2}\). This creates different regions between the function and x-axis: 1. From \(x = -1\) to \(x = \frac{3}{2}\), above the x-axis. 2. From \(x = \frac{3}{2}\) to \(x = 3\), below the x-axis.

Step 4: Calculate area from -1 to 3/2

Calculate the area of the triangle formed from \(x = -1\) to \(x = \frac{3}{2}\). The vertices of this triangle are \((-1, 5), (\frac{3}{2}, 0), (-1, 0)\). The base is from \(x = -1\) to \(x = \frac{3}{2}\), which is \(\frac{5}{2}\), and the height is 5.\[\text{Area}_{1} = \frac{1}{2} \times \frac{5}{2} \times 5 = \frac{25}{4}\]

Step 5: Calculate area from 3/2 to 3

Calculate the area from \(x = \frac{3}{2}\) to \(x = 3\). The vertices are \((\frac{3}{2}, 0), (3, 0), (3, -3)\). This triangle's base is \(3 - \frac{3}{2} = \frac{3}{2}\) and height is 3.\[\text{Area}_{2} = \frac{1}{2} \times \frac{3}{2} \times 3 = \frac{9}{4}\]This area is subtracted (considered negative) because it is below the x-axis.

Step 6: Combine areas

Combine the areas to find the integral:\[\int_{-1}^{3} (3 - 2x) \, dx = \text{Area}_{1} - \text{Area}_{2} = \frac{25}{4} - \frac{9}{4} = \frac{16}{4} = 4\]Therefore, the integral evaluates to 4.

Key Concepts

Linear Function

A linear function is a type of polynomial function in which the highest power of the variable is 1. In simpler terms, a linear function produces a straight line when graphed. For example, the function given in the exercise is \( f(x) = 3 - 2x \). Here, the function is expressed in the form \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) the y-intercept.
The slope, \( m = -2 \), tells us how steep the line is—in this case, the negative sign indicates the line is sloping downwards as it moves from left to right. The y-intercept, \( b = 3 \), is the point where the line crosses the y-axis. This information is key in sketching the function before performing any calculations relating to the area under the line.

Area under a Curve

The concept of the area under a curve involves calculating the space between the curve of a function and the x-axis over a specific interval. For a linear function like \( f(x) = 3 - 2x \), the "curve" is actually a straight line, making it simpler to calculate the area.
In this exercise, we focus on finding the area between the line and the x-axis from \( x = -1 \) to \( x = 3 \). The area is split into two segments because the line crosses the x-axis, leading to sections both above and below it. Understanding how to break down these segments into geometric shapes can simplify calculation, such as viewing the segments as triangles.

Definite Integral

A definite integral represents the net area between a function and the x-axis over a specific interval. It is denoted as \( \int_a^b f(x) \, dx \), where \( a \) and \( b \) are the bounds of integration. When computing definite integrals, both the areas above and beneath the x-axis must be considered.
In our case, the definite integral \( \int_{-1}^{3} (3 - 2x) \, dx \) is used to calculate the entire shaded region between the line \( y = 3 - 2x \) and the x-axis from \( x = -1 \) to \( x = 3 \). Negative area is subtracted, as it falls beneath the x-axis, which affects the net value after summing all areas. Hence, the integral results in 4, indicating a balance of areas above and below the x-axis.

Graphical Interpretation

Graphical interpretation involves visually examining how a function behaves across its domain and how it interacts with the x-axis. This can help in understanding where and how frequently a function is above or below the x-axis, which is crucial when interpreting definite integrals.
By graphing \( y = 3 - 2x \), we can observe the line's intersection points, like the x-intercept at \( x = \frac{3}{2} \). This intercept divides the interval into sections above and below the axis. Visually breaking down these sections can provide insight into calculating net areas, as each portion represents a geometric shape. Such visual aids support verification of calculated areas and enhance comprehension of integral calculus concepts.