one question - spherical polar triple integral with variable change

[cosec(x)]

The question: Evaluate
over the region R which is the upper half of the unit ball, ie.
Using spherical polar coordinates,




So we first work out the bounds in terms of
So, I say




Giving us:








This appears fine to me, but the answers integrate over the region:

So, I say



despite the fact that this defines points for z below the z range, ie. it allows this,
They get an answer different to mine thereby, yet my range of integration seems to make more sense.

Thanks for any help

[enwiabe]

Was this a prac exam question, if so, what year?

[enwiabe]

Your scenario describes the extension of the radius from 0 to 1, with theta wrapping one half of the x-y plane. The reason why you're then getting 0, is that you're going -pi/2 to pi/2. I.E. a negative quadrant to a positive quadrant, and because you're not allowing for symmetry, the negative and positive are cancelling to produce the nullity. Also, for phi, the range is 0 to pi dropping DOWN from the z pole.

I was just discussing the correct way to do this with the esteemed Ahmad (read: he essentially told me how it is )

Phi makes the angle with the vertical, yeah? So to drop down from the vertical z-pole to the x-y plane and not cross it you need an angle of pi/2 only (0, pi/2). To sweep the volume of the ball you then need to rotate around the whole x,y plane 0 to 2pi.

So it should be

0 < r < 1
0 < theta < 2pi
0 < phi < pi/2

I think they (solutions) have their phi and their theta swapped around.

P.S. Suitcase packing is coming along nicely I have a lotttt of material to take in, I've realised.

[cosec(x)]

It was the 2002 semester 1 exam

I still don't see that their ranges sweep it right, it seems that if we are looking at the unit ball with x +ve out of page and z +ve vertical, and y+ve right, it appears that you are sweeping out the front hemisphere.

Why do you allow for z to take a negative value. My region defines the top hemisphere so I integrate over it yet doom befalls me.

So, riddle me this, how come, to get it right you integrate over what looks to be the wrong region?

[enwiabe]

Because, sir cosec( x), they switched phi and theta. You know how in 2D polar co-ords are (r, theta)? They've got it as (r, phi). Hope that settles it for you.

[cosec(x)]

Ahah, I see, who would do that though, honestly everyone uses for angle from the z axis. SPLITTERS!!

I suppose that makes sense then, however can you tell me why my region of integration is wrong, it still specifies the right region, it sweeps out a semicircle and then drags that through a semicircle.

[Ahmad]

is defined for .

[Ahmad]

Ahah, I see, who would do that though, honestly everyone uses for angle from the z axis. SPLITTERS!!

I beg to differ.

[Mao]

why dont we get to do this kind of fun stuff in MUEP?

[enwiabe]

Well if you did UMEP in any year up to 2007 you'd actually have done this stuff It wasn't examinable, but this stuff was on the course.

[Mao]

Well if you did UMEP in any year up to 2007 you'd actually have done this stuff It wasn't examinable, but this stuff was on the course.

thank you, i'll just go harass mum because she didnt give birth to me early enough
cheers,

[enwiabe]

My pleasure

[cosec(x)]

Its not actually on the UMEP course anymore I think, the Melbourne model took what was UMEP and then put it through a linear algebra machine and now its mostly linear algebra, This stuff came from MTH 2010, Multivariable calculus, a second year subject at monash (if you do enhancement you'll do it first semester first year)

[humph]

Its not actually on the UMEP course anymore I think, the Melbourne model took what was UMEP and then put it through a linear algebra machine and now its mostly linear algebra, This stuff came from MTH 2010, Multivariable calculus, a second year subject at monash (if you do enhancement you'll do it first semester first year)

why would they have that in a second year course? it's pretty easy stuff

oh, and i've seen both and as the variable for the z-axis at various times. there's no set standard.

[cosec(x)]

its not about the existence of a set standard, its about the relative coolness of each letter, z is the slightly more obscure axes, as is phi the more obscure letter, thus making them equal on a relative coolness scale.

why would they have that in a second year course? it's pretty easy stuff

The reason it is second year is because at monash, people who don't do spec. maths do this:
year 1 semester 1, MTH1020 (like specialist)
sem. 2 MTH 1030 (first year maths)
second year MTH 2010, multivariable. It is a second year subject simply due to its time of taking place in a maths major.

Anyway, is any maths "hard" if you do it enough (by that I don't mean prove the Taniyama-Shimura Conjecture maths, just routine, discovered yonks ago and institutionalised maths)