Problem 2 - University Calculus: Early Transcendentals

Question

Sketch the described regions of integration. $$-1 \leq x \leq 2, \quad x-1 \leq y \leq x^{2}$$

Step-by-Step Solution

Answer

The region is bounded horizontally by \(-1 \leq x \leq 2\) and vertically between the line \(y = x - 1\) and the curve \(y = x^2\).

Step 1: Understand Region for x

First, analyze the given boundaries for the variable \( x \), which are \( -1 \leq x \leq 2 \). This tells us that the region of integration is horizontally bounded between \( x = -1 \) and \( x = 2 \). Begin by drawing a vertical strip from \( x = -1 \) to \( x = 2 \) on the coordinate plane.

Step 2: Understand Lower Limit for y

Next, analyze the lower boundary for \( y \), which is \( y = x - 1 \). This is a line with a slope of 1 and a y-intercept of -1. Graph this line on the coordinate plane. It will pass through points like \((-1, -2)\) and \((2, 1)\).

Step 3: Understand Upper Limit for y

Then, examine the upper boundary for \( y \), which is \( y = x^2 \). This is a standard parabola opening upward with its vertex at the origin \((0, 0)\). Plot this parabola on the coordinate plane by calculating points like \((-1, 1)\), \((0, 0)\), \((1, 1)\), and \((2, 4)\).

Step 4: Identify Intersections

Identify the points where the line \( y = x - 1 \) intersects with the parabola \( y = x^2 \). Solve the equation \( x - 1 = x^2 \), resulting in \( x^2 - x + 1 = 0 \). Solving gives points of intersection at \( x = 0 \) and \( x = 1 \). Calculate \( y \) for these \( x \) values: \( (0, -1) \) and \((1, 0)\).

Step 5: Sketch Region of Integration

Using the boundaries and intersection points, sketch the region of integration. The region is bounded by the line \( y = x - 1 \) and the parabola \( y = x^2 \) from \( x = -1 \) to \( x = 2 \). Shade the area between these curves from \( x = -1 \) to \( x = 2 \), starting from the line \( y = x - 1 \) (the lower limit) up to the curve \( y = x^2 \) (the upper limit).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional space defined by two perpendicularly intersecting lines called the x-axis and y-axis. It is used for graphically representing mathematical relationships and equations. Each point in this plane is identified by a pair of values

• The first value, often called the x-coordinate, tells you how far to move horizontally from the origin (0,0).
• The second value, known as the y-coordinate, indicates how far to move vertically.

In this problem, both boundaries for x need to be considered before plotting any region. The range of x from -1 to 2 determines the horizontal limits of the area to be sketched. This information helps us establish where on the coordinate plane our region begins and ends horizontally.

Parabolas

A parabola is a symmetrical, U-shaped curve that is described mathematically by a quadratic equation in the form of \( y = ax^2 + bx + c \). For this exercise, the relevant parabola is given by the equation \( y = x^2 \). It opens upwards with the vertex located at the origin, (0, 0). When plotting a parabola on the coordinate plane:

• Find key points such as the vertex and other points like (1, 1), (2, 4), and (-1, 1).
• The line of symmetry for the parabola, in this case, is the y-axis.

The parabola forms the upper boundary of the region of integration in this exercise, influencing how the region is shaped and drawn.

Lines and Slopes

A line in the coordinate plane can be represented by the equation \( y = mx + b \), where m represents the slope, and b the y-intercept. The slope indicates the steepness of the line - a slope of 1 implies that for every unit increase in x, y increases by 1 unit as well.In the line given by \( y = x - 1 \):

• The slope m is 1, indicating this line rises at a 45-degree angle.
• The y-intercept b is -1, which is where the line crosses the y-axis.

This line acts as the lower boundary of the region of integration, and graphical recognition of these characteristics helps establish where the line is plotted on the coordinate plane. Understanding the line's interaction with other curves, like the parabola, relies on knowing its slope and intercept. Points such as (-1, -2) and (2, 1) can be plotted for clarity when sketching.