Another coordinate conversions.

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In summary, you should convert the integral into spherical coordinates, and each x, y, z should go from 0 to \sqrt{1-x^2} or the circle x^2+ y^2= 1.
  • #1
tnutty
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Homework Statement



The actual question is to evaluate the integral. All I need help on is the setting up part.

Instead of making a thread for each, I will post 3 integral question with my attempts.

Just tell me if you see something wrong from rectangular to spherical conversion. Thanks.

1) Evaluate the integral
[tex]\int\int\int_{E} e^\sqrt(x^2+y^2+z^2) dv[/tex]

where E is enclosed by the sphere x^2+y^2+z^2 = 9 in the first octant


This is what I got in spherical coordinate :

[tex]\int^{\pi/2}_{0}[/tex][tex]\int^{\pi/2}_{0}[/tex][tex]\int^{3}_{0} e^\rho\rho^2 *d\rho*sin(\phi)d\phi*d\theta[/tex]



Now for the seconds one :

Evaluate the integral :

[tex]\int\int\int_{E}x^2 dV[/tex]

where E is bounded by the x-z plane and the hemisphere y = sqrt(9-x^2-z^2) and y = sqrt(16 - x^2 - z^2)

Here is my integral setup in spherical coordinates :

[tex]\int^{\pi}_{0}[/tex][tex]\int^{\pi/2}_{0}[/tex][tex]\int^{4}_{3} (p^2*sin^2(\phi)*cos^2(\theta) )* p^2d\phi * sin(\phi) d\phi * d\theta[/tex]

And for the last one :

Evaluate the integral by converting it into spherical coordinate :


In rectangular coordinate :

[tex]\int^{1}_{0}[/tex][tex]\int^{\sqrt(1-x^2}_{0}[/tex][tex]\int^{\sqrt(2-x^2+y^2)}_{\sqrt(x^2+y^2)} xz *dzdydx[/tex]


Here is my partial attempt to convert it into spherical coordinate, but some are missing because I am not sure what it is supposed to be.

[tex]\int^{2*pi}_{0}[/tex] [tex]\int^{?}_{?}[/tex] [tex]\int^{2}_{?} xz \rho^2 d\rho sin(\phi)d\phi d\theta[/tex]

where x = p sin(phi) cos(theta) and z = p*cos(phi)

Thanks for any help.
 
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  • #2
If anyone could help on any of the 3 problems, checking if my conversion is correct, then
I would really appreciate.
 
  • #3
tnutty said:

Homework Statement



The actual question is to evaluate the integral. All I need help on is the setting up part.

Instead of making a thread for each, I will post 3 integral question with my attempts.

Just tell me if you see something wrong from rectangular to spherical conversion. Thanks.

1) Evaluate the integral
[tex]\int\int\int_{E} e^\sqrt(x^2+y^2+z^2) dv[/tex]

where E is enclosed by the sphere x^2+y^2+z^2 = 9 in the first octant


This is what I got in spherical coordinate :

[tex]\int^{\pi/2}_{0}[/tex][tex]\int^{\pi/2}_{0}[/tex][tex]\int^{3}_{0} e^\rho\rho^2 *d\rho*sin(\phi)d\phi*d\theta[/tex]
Yes, that looks good.



Now for the seconds one :

Evaluate the integral :

[tex]\int\int\int_{E}x^2 dV[/tex]

where E is bounded by the x-z plane and the hemisphere y = sqrt(9-x^2-z^2) and y = sqrt(16 - x^2 - z^2)

Here is my integral setup in spherical coordinates :

[tex]\int^{\pi}_{0}[/tex][tex]\int^{\pi/2}_{0}[/tex][tex]\int^{4}_{3} (p^2*sin^2(\phi)*cos^2(\theta) )* p^2d\phi * sin(\phi) d\phi * d\theta[/tex]
Since z includes both positive and negative values, [itex]\phi[/itex] will have to go from 0 to [itex]\pi[/itex].

And for the last one :

Evaluate the integral by converting it into spherical coordinate :


In rectangular coordinate :

[tex]\int^{1}_{0}[/tex][tex]\int^{\sqrt(1-x^2}_{0}[/tex][tex]\int^{\sqrt(2-x^2+y^2)}_{\sqrt(x^2+y^2)} xz *dzdydx[/tex]

Here is my partial attempt to convert it into spherical coordinate, but some are missing because I am not sure what it is supposed to be.

[tex]\int^{2*pi}_{0}[/tex] [tex]\int^{?}_{?}[/tex] [tex]\int^{2}_{?} xz \rho^2 d\rho sin(\phi)d\phi d\theta[/tex]

where x = p sin(phi) cos(theta) and z = p*cos(phi)

Thanks for any help.
Since x ranges from 0 to 1, we are in the right half space. That means that [itex]\theta[/itex] goes from [itex]-\pi/2[/itex] to [itex]\pi/2[/itex], not 0 to [itex]2\pi[/itex]. For each x, y ranges from 0 to [itex]\sqrt{1-x^2}[/itex] or the circle [itex]x^2+ y^2= 1[/itex]. Finally, for each (x,y), z ranges between the cone [itex]z^2= x^2+ y^2[/itex] and the sphere [itex]z^2= 2- x^2-y^2[/itex] (or [itex]x^2+ y^2+ z^2= 2[/itex] which just happen to have the circle [itex]x^2+ y^2= 1[/itex], z= 1 as intersection. In spherical coordinates that cones is [itex]\rho^2cos^2(\phi)= \rho^2 sin^2(\phi)[/itex] or [itex]tan^2(\phi)= 1[/itex], [itex]\phi= \pi/4[/itex] and sphere is [itex]\rho^2 cos^2(\phi)= 2- \rho^2 sin^2(\phi)[/itex] or [itex]\rho^2= 2[/itex], [itex]\rho= \sqrt{2}[/itex]

[itex]\rho[/itex] goes from 0 to [itex]\sqrt{2}[/itex] (the distance from (0,0,0) to each point on the circle of intersection), [itex]\theta[/itex] goes from [itex]-\pi/2[/itex] to [itex]\pi/2[/itex] and [itex]\phi[/itex] goes from 0 to [itex]\pi/4[/itex].
 
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  • #4
There is no way to give reps or something similar, because you earned a million of them.
Thanks a lot, as usual.
 

1. What is the purpose of coordinate conversions?

Coordinate conversions are used to convert the coordinates of a point from one coordinate system to another. This is commonly done in order to accurately represent the location of a point on a map or in a specific reference frame.

2. What are the different types of coordinate systems that can be converted?

There are many different types of coordinate systems that can be converted, including Cartesian coordinates, polar coordinates, geographic coordinates, and projected coordinates. Each system has its own unique way of representing a point's location.

3. How are coordinate conversions performed?

Coordinate conversions are typically performed using mathematical formulas or algorithms that take into account the specific parameters and properties of the two coordinate systems being converted. These formulas can vary based on the type of coordinate system being used.

4. What are some common applications of coordinate conversions?

Coordinate conversions are used in a wide range of fields and applications, including geography, cartography, surveying, navigation, and geocaching. They are also important in computer graphics and image processing for accurately mapping and representing objects in digital space.

5. Are there any limitations or challenges with coordinate conversions?

While coordinate conversions can be highly accurate, there can be some limitations and challenges. One of the main challenges is converting between different coordinate systems that have different scales or units of measurement. Additionally, the accuracy of the conversion can also be affected by factors such as the complexity of the coordinate system and the precision of the input coordinates.

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