Triple Integral Volume question

[ember oyster]

How would I know what order to integrate with respect to? I've posted the question given

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[hoary frost]

you typically integrate inwards-out

so you'd solve this one first

so it is

$(12-4x-4y)-(\frac{3-x-y}{2})$

[earnest crest]

Xenon

[hoary frost]

$(\frac{24-8x-8y}{2})-(\frac{3-x-y}{2})$

[earnest crest]

Xenon

[hoary frost]

$(\frac{21-7x-7y}{2})$

[earnest crest]

Xenon

[hoary frost]

lets call this f(z)

now we can do this

$f(z) = (\frac{21-7x-7y}{2})$

[earnest crest]

Xenon

[hoary frost]

ok, calling it f(z) is wrong, but whatever you get the point

oki

and now

actually this can be written as this

so, i think this is correct, however i might be incorrect, i might have made a mistake somewhere so dont take it for granted, but this is how you would do it

you'd work from the inside and work outwards

my answer is positive

but idk if its correct

[ember oyster]

sorry I should've worded it better😭😭

if I'm only given a region between planes x+y+2z=3 and 4x+4y+z=12 in the first octant(which is what's given in the image), when I'm setting my integrals up, how would I know what order of x, y, or z to integrate in?

like would I integrate it as dz dx dy, dy dz dx, or the other ways to set up the integral?

[hoary frost]

it doesnt matter

the volume is the same no matter which way you integrate it

[ember oyster]

tysm

I thought that was only applicable to rectangles bc of fubinis theorem

[ember oyster]

+close