Triple integral and cartesian coordinates

[queenstudy]

we all know that triple integral can be solved by either cartesian coordinates , spherical ,or cylindrical coordinates
i just need like some advice in knowing when the variable used is constant and when it is not
for example : r in cylindrical coordinates can it be constant or not?? because i always see it in terms of y and z like that
and any advice would be very helpful and thank you

 

[I like Serena]

Hi queenstudy!

Can you give a couple of examples of the integrals you mean?

Taken literally, if you keep r in cylindrical coordinates constant, you have a double integral, meaning you integrate over a cylindrical surface instead of a cylindrical volume.

 

[queenstudy]

Hi queenstudy!

Can you give a couple of examples of the integrals you mean?

Taken literally, if you keep r in cylindrical coordinates constant, you have a double integral, meaning you integrate over a cylindrical surface instead of a cylindrical volume.

by fixing the plane Z=z change the domain of definitoin of the triple integral f(x,y,z) where D is
D=(x,y,z) belongs to R cube and it is the intersection of the sphere and cone (icecream cone) sphere xsquare +y square + z square =2 and z=xsquare + y square
z is positive
in cartesian coordinates
my question here is when i do my projection i will have a circle , and if i want it in cartesian i need like 8 integrals because the plane Z=z should be cut twice and for each domain the intersection is a circle and in each circle we have y and x positive x and y negative and one positive and one negative so can i replace those with r and theta and save time and work and still in cartesian or not??
do you have any extra double and triple integral because i solved on google and want some more and thank you thank you twice serena for helping me out

 

[I like Serena]

Hmm, as yet I'm having trouble understanding what you write...?You appear to refer to the integral (polar):
$$
\int_{r=0}^{r=R} \int_{\theta=0}^{\theta=2\pi} \int_{\phi=0}^{\phi={\pi \over 4}} ... d\phi d\theta dr
$$

Or cartesian:
$$
\int_{x=-\frac R 2 \sqrt 2}^{x=+\frac R 2 \sqrt 2} \int_{y=-\sqrt{R^2-x^2}}^{y=+\sqrt{R^2-x^2}} \int_{z=\sqrt{x^2+y^2}}^{z=\sqrt{R^2-x^2-y^2}} ... dz dy dx
$$Does this look like what you mean?

 

[queenstudy]

what you wrote is true but i am supposed to write z as two constants, and when you study x and y , you must write in terms of constant K , thus in both answers you wrote they arent correct ,
let me ask a simpler question:
i should write the domain in cartesian may i write x and y in polar coordinates and still call them cartesian because if i keep them in cartesian , x and y will have 4 domains when:
1) x and y postive ; 2) x and y negative; 3) x positive and y negative ; 4) x negative and y postive
i hope i explained my question better

 

[I like Serena]

Well, you can rearrange the cartesian integral so z is integrated last.
In that case you would need to split the integral into 2 integrals.
One integral for z from 0 to $\frac R 2 \sqrt 2$ and one from $\frac R 2 \sqrt 2$ to R.
You would not need to split the x or y integrations though.

Perhaps you can show what you think it might be?

Cartesian means not-polar, but why do you think you would have to?
There is no need to split x and y in the 4 domains you mention.

 

[queenstudy]

because according to our proffesor when x is negative y has the possibility to be positive or negative right? so we have two cases
now is x is positive then y can be either positive or negative right? so we also have two possibilities , thus we have 4 cases that's and that is only the case of the intersection of the plane z=constant with the cone
if we do it with the sphere inside the domain we will obtain the same 4 parts thus the total will be 8 parts which is long
that is why we choose polar here for a circle or to me be more specific a full disk because r is only positive and theta is between zero and two pie

 

[I like Serena]

I think you're mixing up a single integral with higher dimensional integrals.

If you have a single integral to find the area of (part of) a circle disk, you have the problem that y has a negative and positive value and you have to choose between one of them.

If you have a double integral over a circle disk, y runs from the negative value to the positive value, which covers the entire circle disk.
So there is no need to split up the integral.

 

[queenstudy]

but in class when i we projected the domain we got a circle , and it got 4 domains where x and y contain the domain of the circle
okay if we have a disk on the xoy axis and i want to make double integration with respect to x and y don't i get 4 parts of the circles thus 4 domains?? then we use polar coordinate and becomes a simple integration right?
if this is true , MY question is that can r and theta be considered also as cartesian coordinates?

 

[I like Serena]

Can you show me integrals of what you mean?

As for your question whether r and theta can be considered cartesian coordinates, I believe we covered that in your other thread.
All I can say is that the word "cartesian" means x and y, which is not r and theta.