Problem 55 - Calculus for Biology and Medicine

Question

Use a graph to interpret the definite integral in terms of areas. Do not compute the integrals. $$ \int_{-1}^{2}\left(x^{2}-1\right) d x $$

Step-by-Step Solution

Answer

The integral represents the total net area between the graph and the x-axis from \( x = -1 \) to \( x = 2 \), considering positive and negative areas.

Step 1: Understanding the Problem

The exercise requires interpreting the definite integral \( \int_{-1}^{2}(x^2-1) \, dx \) in terms of areas under the curve of the function \( f(x) = x^2 - 1 \) from \( x = -1 \) to \( x = 2 \). Instead of computing the integral, we will visualize and interpret it using a graph.

Step 2: Sketch the Function

Begin by sketching the graph of the function \( f(x) = x^2 - 1 \). This function is a parabola opening upwards with its vertex at \( (0, -1) \), since it is a vertical shift of the basic parabola \( y = x^2 \) down by 1 unit.

Step 3: Identify the Region of Interest

The integration limits \( x = -1 \) and \( x = 2 \) define the interval of interest. Mark these points on the \(x\)-axis and draw vertical lines up to the graph of the function at each of these \(x\) values. This helps visualize the area we need to interpret.

Step 4: Find Points of Intersection with x-axis

Calculate the x-intercepts (where \( f(x) = 0 \)) to determine where the parabola intersects the x-axis. Solve \( x^2 - 1 = 0 \) to find \( x = 1 \) and \( x = -1 \). These are the points where the graph crosses the x-axis.

Step 5: Partition the Area

Divide the area into sections. In this case, consider two parts: one from \( x = -1 \) to \( x = 1 \), and the other from \( x = 1 \) to \( x = 2 \). The first part is symmetric and includes some area above the x-axis and some below, while the second part is entirely above the x-axis.

Step 6: Interpret Area Sections

The integral represents the net area: positive above the x-axis and negative below. From \( x = -1 \) to \( x = 1 \), the areas above and below the x-axis cancel each other partially, while from \( x = 1 \) to \( x = 2 \), the entire area is above the x-axis, contributing positively.

Key Concepts

Area under Curve

The definite integral can be visualized as the net area under the curve between two points on the x-axis, specifically from \( x = -1 \) to \( x = 2 \) for this exercise. Imagine a two-dimensional space where the curve of a function, such as \( y = x^2 - 1 \), hovers over a line or crosses it. This curve essentially partitions the plane into sections of negative and positive areas with respect to the x-axis.
The net area accounts for everything:

• The areas above the x-axis count as positive.
• The areas below the x-axis count as negative.

In our exercise, the curve dissects the plane generating these regions. Between \( x = -1 \) and \( x = 1 \), the curve goes below the x-axis, while between \( x = 1 \) and \( x = 2 \), it stays above. By understanding this division, you can infer the overall magnitude and direction of the integral, indicating the cumulative effect of heights of the curve along the interval.

Graph Interpretation

Interpreting a graph means discerning the underlying story it tells about a function. For a function like \( f(x) = x^2 - 1 \), the graph is a parabola. It starts below the line \( y=0 \) or the x-axis at its vertex point \( (0, -1) \), then rises as it moves away from the vertex. Graph interpretation requires you to visually

• Identify intercepts, where the curve crosses axes.
• Locate regions of interest based on integration limits, portraying a visual map of the net area we calculate.

By marking the interval from \( x = -1 \) to \( x = 2 \) on the graph, the areas derived indicate different behavior of the function. The curves' intersections at the points, where \( x = -1 \) and \( x = 1 \), are crucial as they help delineate segments that contribute to net area positively or negatively. Reading these characteristics affords vital insights into the functions' nature and transformations over the given range.

Functions and Graphs

The function in question, \( f(x) = x^2 - 1 \), is a quadratic function, characterized by a parabolic shape. Functions and their corresponding graphs paint a picture of their behavior, allowing transformations from algebraic form to visual understanding. Here's how to decode this:

• The basic form \( y=x^2 \) represents a standard parabola opening upward, centered at the origin \( (0, 0) \).
• The transformation \( f(x) = x^2 - 1 \) shifts this parabola downward by one unit due to the \(-1\).
• Analyze the symmetry and axes of the function, especially the y-axis as the line of symmetry for parabolic forms.

Graphs are visual language, acting as windows to functions' behaviors. Plotting \( x = -1 \) to \( x = 2 \) shows intercepts, the height shift, and how to discern between positive and negative areas related to the x-axis. Understanding graphs isn't just about seeing shapes; it's about connecting algebraic ideas with visual methods to enhance comprehension.