Volume Integrals of a sphere

[Davio]

Hey guys, could one of you explain why when doing a volume integral using spherical polar coordinates, you have the limits as 2 pi to 0 on phi but only pi to 0 on theta? Thanks.

To clarify, I've been doing this all this time for questions, but it just occurred to me that I Don't know why i do that .

 

[tiny-tim]

Hi Davio!

(Of course, you can't go from o to 2π on both, because then you'd be covering every point twice!)

Well, you can always do it the other way round … but the integrals that usually occur in practice just happen to be symmetric in phi, so integrating from 0 to 2π on phi is dead easy!

In particular, if you're converting from (x,y,z), then you get ∫∫∫(r^2)sinthetadrdthetadphi … and that itself is symmetric in phi, so with luck the whole thing immediately becomes 2π∫∫r^2)sinthetadrdtheta.

 

[Davio]

Hmmm, but wouldn't it be, for both phi and theta, just pi to zero - ie. half each? Or am imagining this wrong . R is radius, so integrating along that place, in a circle, theta and phi are both angles, just in different directions? My maths is a bit poor (not good for a physics major)!

 

[arildno]

Hi, Davio!

Take a line segment from the centre of the ball to its surface. That line segment makes the angle theta with the "z"-axis.

In order to reach ALL points on the circle with the same angle to the "xy"-plane, you rotate the line segment around the z-axis, with phi then going from 0 to 2pi.

Now, you will have covered ALL these circles (and hence all points) for theta-values going from 0 (i.e, the line segment runs along the positive z-half axis) to pi, (i.e along the negative half axis for z)

 

[tiny-tim]

… don't ignore America …

Yup, arildno is right!

Start with a very long string, fixed at the North pole. Take the other end down the Greenwich meridian from theta = 0 to π. Now you're at the South pole.

So far so good …

Now sweep the string round from phi = 0 to π. You'll cover most of Europe and Africa and the whole of Asia and Australia.

But you'll stop at the International Date Line!

What about America? ​

 

[mathman]

One simple way to see it. Latitude goes from -90 to 90, while longitude goes from -180 to 180.

 

[HallsofIvy]

It's those blasted physicists again! They keep swapping $\theta$ and $\phi$ on us!

 

[Davio]

I'm going to sit down and think about the replies in a minute, I'm on to a question about conical cones now, does anyone have any good resources for understand the images behind integration? I can integrate etc, but can't quite understand the limits of weird shapes, or even normal shapes!

 

[tiny-tim]

… painting problem …

Hi Davio!

Do you mean comical cones?

Now, they are weird!

But keep this in perspective … this isn't an integration problem … it's only a painting problem.

Imagine you have to program a robot to paint a sphere … do you tell it to paint from 0 to π, or 2π, for each coordinate?

If you give it the wrong instructions, it'll either waste paint or not use enough!

That's your only problem … making sure that everything is covered which should be!