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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 2 , a point with rectangular coordinates ( x , y ) can be identified with ( r , θ ) in polar coordinates and vice versa, where x = r cos θ , y = r sin θ , r 2 = x 2 + y 2 and tan θ = ( y x ) are the relationships between the variables.

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

In xyz space, a point is shown (x, y, z). There is also a depiction of it in polar coordinates as (r, theta, z).
Cylindrical coordinates are similar to polar coordinates with a vertical z coordinate added.

To convert from rectangular to cylindrical coordinates, we use the conversion x = r cos θ and y = r sin θ . To convert from cylindrical to rectangular coordinates, we use r 2 = x 2 + y 2 and θ = tan −1 ( y x ) . 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 θ = α (constant) we mean an infinite ray making an angle α with the positive x -axis.

Similarly, in three-dimensional space with rectangular coordinates ( x , y , z ) , the equations x = k , y = l , and z = m , where k , l , and m are constants, represent unbounded planes parallel to the y z -plane, x z -plane and x y -plane, respectively. With cylindrical coordinates ( r , θ , z ) , by r = c , θ = α , and z = m , where c , α , and m are constants, we mean an unbounded vertical cylinder with the z -axis as its radial axis; a plane making a constant angle α with the x y -plane; and an unbounded horizontal plane parallel to the x y -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.)

Questions & Answers

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.
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
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How can I make nanorobot?
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Smarajit Reply
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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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