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Chapter 12: Problem 34

Give the equation of a curve in one of the coordinate planes. Write an equation for the surface generated by revolving this curve around the indicated axis. Then sketch the surface. \(z=4-x^{2} ;\) the \(z\) -axis

Short Answer

Expert verified
The surface is a paraboloid: \( z = 4 - (x^2 + y^2) \).

Step by step solution

01

Identify the Curve

First, identify the given curve equation. The equation provided is for the curve in the x-z coordinate plane: \( z = 4 - x^2 \). This is a parabola opening downwards.
02

Understand the Surface Revolution

The task is to revolve this curve around the z-axis to create a surface. Revolving a curve around an axis results in a 3D surface.
03

Introduce Cylindrical Coordinates

Since we're revolving around the z-axis, we will use cylindrical coordinates where \( x = r \cos \theta \) and \( y = r \sin \theta \). This helps in describing the radius (\(r\)) at any angle (\(\theta\)) around the z-axis.
04

Set Up the Surface Equation

Replace \( x \) with \( r \) in the original equation \( z = 4 - x^2 \), as the distance from the z-axis is the radius (\( r \)):\[ z = 4 - r^2 \].
05

Express in Cartesian Coordinates

To express this surface using Cartesian coordinates x and y, substitute \( r = \sqrt{x^2 + y^2} \) into the equation: \[ z = 4 - (x^2 + y^2) \]. This equation describes the paraboloid resulting from revolving the curve around the z-axis.
06

Sketch the Surface

Visualize the surface. This revolution around the z-axis forms a paraboloid, which looks like a 3D bowl opening downwards with its vertex at the point (0, 0, 4) on the z-axis.

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Key Concepts

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

Cylindrical Coordinates
Cylindrical coordinates offer a way to describe points in a three-dimensional space using a combination of polar coordinates in the xy-plane and height along the z-axis. This system is particularly useful when dealing with problems involving symmetry around the z-axis, such as surfaces of revolution.
  • The cylindrical coordinate system uses three parameters: the radius \( r \), the angle \( \theta \), and the height \( z \).
  • These coordinates relate to Cartesian coordinates \( (x, y, z) \) by the equations: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( z = z \).
  • The radius \( r \) represents the distance of a point from the z-axis in the xy-plane.
  • The angle \( \theta \) indicates the direction of the point from the positive x-axis.
Using cylindrical coordinates simplifies the process of solving problems involving rotation around the z-axis because it aligns naturally with circular and radial features.
Paraboloid
A paraboloid is a type of surface that can be described as a 3-dimensional counterpart to a parabola. When a parabolic curve is revolved around its axis of symmetry, the resulting surface is a paraboloid.
  • In our exercise, the parabola \( z = 4 - x^2 \) is rotated around the z-axis.
  • This process creates a 3D structure that looks like a bowl or a dish with a central peak or vertex.
  • The equation after revolving becomes \( z = 4 - (x^2 + y^2) \), representing a circular paraboloid oriented along the z-axis.
  • This paraboloid has its vertex at the point \((0, 0, 4)\) and opens downwards, forming a shape symmetric about the z-axis.
Understanding paraboloids is important in various fields, from designing satellite dishes and reflecting telescopes to optimizing structures in architecture.
Curve Equation
When discussing curve equations within different coordinate systems, it's vital to understand how they represent shapes within a plane. In this instance, the provided curve equation is \( z = 4 - x^2 \), a simple expression representing a parabola in the xz-plane.
  • The equation tells us the relationship between the variables \( x \) and \( z \). As \( x \) varies, \( z \) describes how high or low the curve points are.
  • With \( z = 4 - x^2 \), the highest point of the parabola is at the vertex (0, 4) in the xz-plane, and it opens downwards since the square of \( x \) is subtracted from 4.
  • Recognizing this curve as a parabola helps in visualizing how revolving it around an axis generates a 3D surface, thus aiding in the transition to a surface of revolution.
Grasping curve equations is foundational for comprehending more complex mathematical models and physical phenomena.
Coordinate Planes
Coordinate planes are the reference planes you'd encounter in three-dimensional Cartesian coordinates. Each plane is defined by two of the three axes (x, y, z).
  • The xy-plane is horizontal, spanning the x and y axes with \( z = 0 \).
  • The xz-plane involves the x and z axes with \( y = 0 \). This is where our original curve \( z = 4 - x^2 \) is situated.
  • The yz-plane involves the y and z axes with \( x = 0 \).
Understanding these planes is crucial because mathematical problems often refer to coordinates or transformations involving these reference planes. By visualizing these planes and their intersections, one can better interpret equations and sketches of surfaces, aiding in the comprehension of spatial geometries and functions.

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Most popular questions from this chapter

Prove that the projection into the \(x y\) -plane of the intersection of the plane \(x+y+z=1\) and the ellipsoid \(x^{2}+4 y^{2}+4 z^{2}=4\) is an ellipse.

Describe and sketch the graphs of the equations given in Problems 1 through 30 . \(x^{2}+y^{2}-9 z^{2}=9\)

Determine whether the graph (in the \(x y\) -plane) of the given equation is an ellipse or a hyperbola. Check your answer graphically if you have access to a computer algebra system with a "contour plotting" facility. \(40 x^{2}+36 x y+25 y^{2}=101\)

Determine whether the graph of the given equation is an elliptic or a hyperbolic paraboloid. Check your answer graphically by plotting the surface. \(z=4 x^{2}+6 x y-4 y^{2}\)

Find both the cylindrical coordinates and the spherical coordinates of the point \(P\) with the given rectangular coordinates. \(P(2,1,-2)\)
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