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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. 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] ). 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. $y=\text{sin}\phantom{\rule{0.2em}{0ex}}x$
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). 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).
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 $y=\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}x$ (see the following figure). 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.

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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