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Chapter 14: Problem 2

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \((2, 1)\).
03

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 \((2, 4)\).
04

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 \((1, 0)\).
05

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).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A solid ball is bounded by the sphere \(\rho=a\) Find the moment of inertia about the \(z\) -axis if the density is a. \(\delta(\rho, \phi, \theta)=\rho^{2}\) b. \(\delta(\rho, \phi, \theta)=r=\rho \sin \phi\)

A solid is bounded on the top by the paraboloid \(z=r^{2},\) on the bottom by the plane \(z=0,\) and on the sides by the cylinder \(r=1 .\) Find the center of mass and the moment of inertia about the \(z\) -axis if the density is a. \(\delta(r, \theta, z)=z\) b. \(\delta(r, \theta, z)=r\)

A spherical planet of radius \(R\) has an atmosphere whose density is \(\mu=\mu_{0} e^{-c h},\) where \(h\) is the altitude above the surface of the planet, \(\mu_{0}\) is the density at sea level, and \(c\) is a positive constant. Find the mass of the planet's atmosphere.

In Exercises \(37-40,\) find the average value of \(F(x, y, z)\) over the given region. \(F(x, y, z)=x^{2}+y^{2}+z^{2}\) over the cube in the first octant bounded by the coordinate planes and the planes \(x=1, y=1\) and \(z=1\).

Find the volumes of the regions in Exercises \(23-36\). The region between the cylinder \(z=y^{2}\) and the \(x y\) -plane that is bounded by the planes \(x=0, x=1, y=-1, y=1\) (GRAPH CAN'T COPY).
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