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  • Identify a cylinder as a type of three-dimensional surface.
  • Recognize the main features of ellipsoids, paraboloids, and hyperboloids.
  • Use traces to draw the intersections of quadric surfaces with the coordinate planes.

We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces , to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.

Identifying cylinders

The first surface we’ll examine is the cylinder. Although most people immediately think of a hollow pipe or a soda straw when they hear the word cylinder , here we use the broad mathematical meaning of the term. As we have seen, cylindrical surfaces don’t have to be circular. A rectangular heating duct is a cylinder, as is a rolled-up yoga mat, the cross-section of which is a spiral shape.

In the two-dimensional coordinate plane, the equation x 2 + y 2 = 9 describes a circle centered at the origin with radius 3 . In three-dimensional space, this same equation represents a surface. Imagine copies of a circle stacked on top of each other centered on the z -axis ( [link] ), forming a hollow tube. We can then construct a cylinder from the set of lines parallel to the z -axis passing through circle x 2 + y 2 = 9 in the xy -plane, as shown in the figure. In this way, any curve in one of the coordinate planes can be extended to become a surface.

This figure a 3-dimensional coordinate system. It has a right circular center with the z-axis through the center. The cylinder also has points labeled on the x and y axis at (3, 0, 0) and (0, 3, 0).
In three-dimensional space, the graph of equation x 2 + y 2 = 9 is a cylinder with radius 3 centered on the z -axis. It continues indefinitely in the positive and negative directions.

Definition

A set of lines parallel to a given line passing through a given curve is known as a cylindrical surface, or cylinder    . The parallel lines are called rulings    .

From this definition, we can see that we still have a cylinder in three-dimensional space, even if the curve is not a circle. Any curve can form a cylinder, and the rulings that compose the cylinder may be parallel to any given line ( [link] ).

This figure has a 3-dimensional surface that begins on the y-axis and curves upward. There is also the x and z axes labeled.
In three-dimensional space, the graph of equation z = x 3 is a cylinder, or a cylindrical surface with rulings parallel to the y -axis.

Graphing cylindrical surfaces

Sketch the graphs of the following cylindrical surfaces.

  1. x 2 + z 2 = 25
  2. z = 2 x 2 y
  3. y = sin x
  1. The variable y can take on any value without limit. Therefore, the lines ruling this surface are parallel to the y -axis. The intersection of this surface with the xz -plane forms a circle centered at the origin with radius 5 (see the following figure).
    This figure is the 3-dimensional coordinate system. It has a right circular cylinder on its side with the y-axis in the center. The cylinder intersects the x-axis at (5, 0, 0). It also has two points of intersection labeled on the z-axis at (0, 0, 5) and (0, 0, -5).
    The graph of equation x 2 + z 2 = 25 is a cylinder with radius 5 centered on the y -axis.
  2. In this case, the equation contains all three variables x , y , and z so none of the variables can vary arbitrarily. The easiest way to visualize this surface is to use a computer graphing utility (see the following figure).
    This figure has a surface in the first octant. The cross section of the solid is a parabola.
  3. In this equation, the variable z can take on any value without limit. Therefore, the lines composing this surface are parallel to the z -axis. The intersection of this surface with the yz -plane outlines curve y = sin x (see the following figure).
    This figure is a three dimensional surface. A cross section of the surface parallel to the x y plane would be the sine curve.
    The graph of equation y = sin x is formed by a set of lines parallel to the z -axis passing through curve y = sin x in the xy -plane.
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Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Good
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
Practice Key Terms 9

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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