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Chapter 14: Problem 54

Find an equation of the surface obtained by revolving the graph of the equation about the indicated axis. $$z=e^{-y^{2}} ; \quad y-axis$$

Short Answer

Expert verified
The equation is \( x^2 + z^2 = e^{-2y^2} \).

Step by step solution

01

Understanding the Problem

We need to find the equation of the surface formed by revolving the given function around the y-axis. The function given is expressed in terms of z and y, specifically, \( z = e^{-y^{2}} \).
02

Visualizing Surface Revolution

When a curve is revolved around the y-axis, the resulting shape is a surface of revolution. Each point \((y, z)\) in the original graph traces out a circle in the xz-plane.
03

Setting Up the Cylindrical Coordinates

In cylindrical coordinates, a surface of revolution around the y-axis is represented by \((x^2 + z^2 = r^2)\), where \(r\) is a radius function of \(y\). For the given function, \(r = e^{-y^2}\).
04

Formulating the Surface Equation

To represent this surface, we use the relation that the radius from the y-axis (\(r\)) is expressed as \( \sqrt{x^2 + z^2} = e^{-y^2} \).
05

Simplifying the Equation

Squaring both sides gives us the equation \( x^2 + z^2 = e^{-2y^2} \). This is the equation of the surface obtained by revolving the given function around the y-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 provide a blend of cartesian and polar coordinate systems, creating an efficient way to describe 3D shapes. In this system, any point in space is defined through three variables:
  • \(r\): the radial distance from the chosen axis (typically the z-axis in physics, but here we are revolving around the y-axis).
  • \(\theta\): the angular coordinate, specifying the angle between the x-axis and the line connecting the point to the origin.
  • \(y\): the height, or position along the y-axis.
This configuration can be visualized as generalizing the concept of polar coordinates into three dimensions. It's particularly useful for problems involving symmetry around a central axis. When we use cylindrical coordinates, calculations for volumes and surface areas of curved 3D shapes become much simpler. Knowing how to switch between Cartesian and cylindrical coordinates is essential for understanding surfaces such as those created by revolution.
Revolving Around an Axis
Revolving a shape around an axis transforms a 2D curve into a 3D surface. Imagine spinning a curve as if it's being rotated in a circle, capturing all the positions each point on the curve could take. This is the process of revolution. In the problem we have, the graph of the function \(z = e^{-y^2}\) is revolved around the y-axis.
  • The original curve is in the yz-plane.
  • As each point on the curve makes a complete rotation, it traces out a circle in the xz-plane.
  • These circles stack vertically along the y-axis, forming the entire surface.
In essence, revolution allows us to take a planar shape and stretch it into a complex geometric structure.
Equation of a Surface
To describe a surface formed by revolving a function, a new equation that accounts for three dimensions is necessary. The original function is just part of this process. In the given exercise, we started with the equation \(z = e^{-y^2}\). When revolved around the y-axis, every point turns into a circle of radius equal to the z-value at that particular y.
  • The radius function becomes \(e^{-y^2}\).
  • Each circle's equation combines with z, forming the surface equation.
  • From cylindrical coordinates, the surface equation is \(x^2 + z^2 = e^{-2y^2}\).
Thus, what began as a simple 2D function is transformed into a 3D surface equation. These equations reveal how rotational symmetry about an axis imparts unique properties to the surface.

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Most popular questions from this chapter

Find parametric equations for the line of intersection of the two planes. $$ x+2 y-9 z=7, \quad 2 x-3 y+17 z=0 $$

Sketch the graph of the equation in an \(x y z-\) coordinate system, and identify the surface. $$2 x+4 y+3 z=12$$

Describe the region \(R\) in a three-dimensional coordinate system. $$ R=\left\\{(x, y, z):\left(x^{2} / 4\right)+\left(y^{2} / 9\right) \geq 1\right\\} $$

Sketch the graph of the quadric surface. Paraboloids (a) \(z=x^{2}+\frac{y^{2}}{9}\) (b) \(\frac{z^{2}}{25}+\frac{y^{2}}{9}-x=0\)

Sketch the graph of the equation in an \(x y z-\) coordinate system, and identify the surface. $$z=e^{y}$$
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