Having issues understanding the notion of 2D volume in Measure Theory.

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So the exercise I'm trying to solve goes like this:
P ⊂ $R^2$ is the interior of a parallelogram defined by its' corners coordinates:
(-2,1), (-1,3), (-3,3), (-4,1)
I need to find out $vol_2 (P)$



Now I found somewhere in a book that if the set I'm looking at is



$P=[a_1,b_1] times [a_2,b_2] times ... times [a_n,b_n]$ ⊂ $R^n$



then



$vol_n (P)$=$|b_1-a_1| |b_1-a_1|...|b_n-a_n|$



And I believe this finding may prove to be useful in what I need to solve, but I fail to identify the terms $[a_1,b_1]$ and $[a_2,b_2]$ in my scenario. Also, I don't really understand the concept of having a volume if this is a 2 dimensional figure.







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  • Your formula won't work, because it's not a 2 dimensional cube. Also, specify your question. What options do you have to evaluate the area. You can obviously just use elementary school formulas, but I guess that's a no?
    – Rumpelstiltskin
    Aug 8 at 21:34











  • The author's note was that the formula is for any n-dimensional parallelepiped. Yeah I'm not supposed to use elementary school geometry, but measure theory and the likes of Lebesgue integration.
    – Kira HD
    Aug 8 at 21:37











  • I guess you can just integrate $1$ over your set? Would that do? Your formula doesn't exactly work here, it's not of the form like $P$
    – Rumpelstiltskin
    Aug 8 at 21:43











  • I have no idea sadly..
    – Kira HD
    Aug 8 at 21:44










  • Did you cover double integrals?
    – Rumpelstiltskin
    Aug 8 at 21:44














up vote
0
down vote

favorite












So the exercise I'm trying to solve goes like this:
P ⊂ $R^2$ is the interior of a parallelogram defined by its' corners coordinates:
(-2,1), (-1,3), (-3,3), (-4,1)
I need to find out $vol_2 (P)$



Now I found somewhere in a book that if the set I'm looking at is



$P=[a_1,b_1] times [a_2,b_2] times ... times [a_n,b_n]$ ⊂ $R^n$



then



$vol_n (P)$=$|b_1-a_1| |b_1-a_1|...|b_n-a_n|$



And I believe this finding may prove to be useful in what I need to solve, but I fail to identify the terms $[a_1,b_1]$ and $[a_2,b_2]$ in my scenario. Also, I don't really understand the concept of having a volume if this is a 2 dimensional figure.







share|cite|improve this question




















  • Your formula won't work, because it's not a 2 dimensional cube. Also, specify your question. What options do you have to evaluate the area. You can obviously just use elementary school formulas, but I guess that's a no?
    – Rumpelstiltskin
    Aug 8 at 21:34











  • The author's note was that the formula is for any n-dimensional parallelepiped. Yeah I'm not supposed to use elementary school geometry, but measure theory and the likes of Lebesgue integration.
    – Kira HD
    Aug 8 at 21:37











  • I guess you can just integrate $1$ over your set? Would that do? Your formula doesn't exactly work here, it's not of the form like $P$
    – Rumpelstiltskin
    Aug 8 at 21:43











  • I have no idea sadly..
    – Kira HD
    Aug 8 at 21:44










  • Did you cover double integrals?
    – Rumpelstiltskin
    Aug 8 at 21:44












up vote
0
down vote

favorite









up vote
0
down vote

favorite











So the exercise I'm trying to solve goes like this:
P ⊂ $R^2$ is the interior of a parallelogram defined by its' corners coordinates:
(-2,1), (-1,3), (-3,3), (-4,1)
I need to find out $vol_2 (P)$



Now I found somewhere in a book that if the set I'm looking at is



$P=[a_1,b_1] times [a_2,b_2] times ... times [a_n,b_n]$ ⊂ $R^n$



then



$vol_n (P)$=$|b_1-a_1| |b_1-a_1|...|b_n-a_n|$



And I believe this finding may prove to be useful in what I need to solve, but I fail to identify the terms $[a_1,b_1]$ and $[a_2,b_2]$ in my scenario. Also, I don't really understand the concept of having a volume if this is a 2 dimensional figure.







share|cite|improve this question












So the exercise I'm trying to solve goes like this:
P ⊂ $R^2$ is the interior of a parallelogram defined by its' corners coordinates:
(-2,1), (-1,3), (-3,3), (-4,1)
I need to find out $vol_2 (P)$



Now I found somewhere in a book that if the set I'm looking at is



$P=[a_1,b_1] times [a_2,b_2] times ... times [a_n,b_n]$ ⊂ $R^n$



then



$vol_n (P)$=$|b_1-a_1| |b_1-a_1|...|b_n-a_n|$



And I believe this finding may prove to be useful in what I need to solve, but I fail to identify the terms $[a_1,b_1]$ and $[a_2,b_2]$ in my scenario. Also, I don't really understand the concept of having a volume if this is a 2 dimensional figure.









share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 8 at 21:32









Kira HD

143




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  • Your formula won't work, because it's not a 2 dimensional cube. Also, specify your question. What options do you have to evaluate the area. You can obviously just use elementary school formulas, but I guess that's a no?
    – Rumpelstiltskin
    Aug 8 at 21:34











  • The author's note was that the formula is for any n-dimensional parallelepiped. Yeah I'm not supposed to use elementary school geometry, but measure theory and the likes of Lebesgue integration.
    – Kira HD
    Aug 8 at 21:37











  • I guess you can just integrate $1$ over your set? Would that do? Your formula doesn't exactly work here, it's not of the form like $P$
    – Rumpelstiltskin
    Aug 8 at 21:43











  • I have no idea sadly..
    – Kira HD
    Aug 8 at 21:44










  • Did you cover double integrals?
    – Rumpelstiltskin
    Aug 8 at 21:44
















  • Your formula won't work, because it's not a 2 dimensional cube. Also, specify your question. What options do you have to evaluate the area. You can obviously just use elementary school formulas, but I guess that's a no?
    – Rumpelstiltskin
    Aug 8 at 21:34











  • The author's note was that the formula is for any n-dimensional parallelepiped. Yeah I'm not supposed to use elementary school geometry, but measure theory and the likes of Lebesgue integration.
    – Kira HD
    Aug 8 at 21:37











  • I guess you can just integrate $1$ over your set? Would that do? Your formula doesn't exactly work here, it's not of the form like $P$
    – Rumpelstiltskin
    Aug 8 at 21:43











  • I have no idea sadly..
    – Kira HD
    Aug 8 at 21:44










  • Did you cover double integrals?
    – Rumpelstiltskin
    Aug 8 at 21:44















Your formula won't work, because it's not a 2 dimensional cube. Also, specify your question. What options do you have to evaluate the area. You can obviously just use elementary school formulas, but I guess that's a no?
– Rumpelstiltskin
Aug 8 at 21:34





Your formula won't work, because it's not a 2 dimensional cube. Also, specify your question. What options do you have to evaluate the area. You can obviously just use elementary school formulas, but I guess that's a no?
– Rumpelstiltskin
Aug 8 at 21:34













The author's note was that the formula is for any n-dimensional parallelepiped. Yeah I'm not supposed to use elementary school geometry, but measure theory and the likes of Lebesgue integration.
– Kira HD
Aug 8 at 21:37





The author's note was that the formula is for any n-dimensional parallelepiped. Yeah I'm not supposed to use elementary school geometry, but measure theory and the likes of Lebesgue integration.
– Kira HD
Aug 8 at 21:37













I guess you can just integrate $1$ over your set? Would that do? Your formula doesn't exactly work here, it's not of the form like $P$
– Rumpelstiltskin
Aug 8 at 21:43





I guess you can just integrate $1$ over your set? Would that do? Your formula doesn't exactly work here, it's not of the form like $P$
– Rumpelstiltskin
Aug 8 at 21:43













I have no idea sadly..
– Kira HD
Aug 8 at 21:44




I have no idea sadly..
– Kira HD
Aug 8 at 21:44












Did you cover double integrals?
– Rumpelstiltskin
Aug 8 at 21:44




Did you cover double integrals?
– Rumpelstiltskin
Aug 8 at 21:44















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