# 5.5 Triple integrals in cylindrical and spherical coordinates

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• Evaluate a triple integral by changing to cylindrical coordinates.
• Evaluate a triple integral by changing to spherical coordinates.

Earlier in this chapter we showed how to convert a double integral in rectangular coordinates into a double integral in polar coordinates in order to deal more conveniently with problems involving circular symmetry. A similar situation occurs with triple integrals, but here we need to distinguish between cylindrical symmetry and spherical symmetry. In this section we convert triple integrals in rectangular coordinates into a triple integral in either cylindrical or spherical coordinates.

Also recall the chapter opener, which showed the opera house l’Hemisphèric in Valencia, Spain. It has four sections with one of the sections being a theater in a five-story-high sphere (ball) under an oval roof as long as a football field. Inside is an IMAX screen that changes the sphere into a planetarium with a sky full of $9000$ twinkling stars. Using triple integrals in spherical coordinates, we can find the volumes of different geometric shapes like these.

## Review of cylindrical coordinates

As we have seen earlier, in two-dimensional space ${ℝ}^{2},$ a point with rectangular coordinates $\left(x,y\right)$ can be identified with $\left(r,\theta \right)$ in polar coordinates and vice versa, where $x=r\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,$ $y=r\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ,$ ${r}^{2}={x}^{2}+{y}^{2}$ and $\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =\left(\frac{y}{x}\right)$ are the relationships between the variables.

In three-dimensional space ${ℝ}^{3},$ a point with rectangular coordinates $\left(x,y,z\right)$ can be identified with cylindrical coordinates $\left(r,\theta ,z\right)$ and vice versa. We can use these same conversion relationships, adding $z$ as the vertical distance to the point from the $xy$ -plane as shown in the following figure.

To convert from rectangular to cylindrical coordinates, we use the conversion $x=r\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$ and $y=r\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta .$ To convert from cylindrical to rectangular coordinates, we use ${r}^{2}={x}^{2}+{y}^{2}$ and $\theta ={\text{tan}}^{-1}\left(\frac{y}{x}\right).$ The $z$ -coordinate remains the same in both cases.

In the two-dimensional plane with a rectangular coordinate system, when we say $x=k$ (constant) we mean an unbounded vertical line parallel to the $y$ -axis and when $y=l$ (constant) we mean an unbounded horizontal line parallel to the $x$ -axis. With the polar coordinate system, when we say $r=c$ (constant), we mean a circle of radius $c$ units and when $\theta =\alpha$ (constant) we mean an infinite ray making an angle $\alpha$ with the positive $x$ -axis.

Similarly, in three-dimensional space with rectangular coordinates $\left(x,y,z\right),$ the equations $x=k,y=l,$ and $z=m,$ where $k,l,$ and $m$ are constants, represent unbounded planes parallel to the $yz$ -plane, $xz$ -plane and $xy$ -plane, respectively. With cylindrical coordinates $\left(r,\theta ,z\right),$ by $r=c,\theta =\alpha ,$ and $z=m,$ where $c,\alpha ,$ and $m$ are constants, we mean an unbounded vertical cylinder with the $z$ -axis as its radial axis; a plane making a constant angle $\alpha$ with the $xy$ -plane; and an unbounded horizontal plane parallel to the $xy$ -plane, respectively. This means that the circular cylinder ${x}^{2}+{y}^{2}={c}^{2}$ in rectangular coordinates can be represented simply as $r=c$ in cylindrical coordinates. (Refer to Cylindrical and Spherical Coordinates for more review.)

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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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?