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Double Integrals.

So I've been learning how to do double integrals such as:

∬R (xy+4) dy dx \iint_{R}^{~} (xy+4)~dy~dx

But I'm not actually sure what this means, or represents geometrically speaking.

Anyone want to clarify what this operation is actually doing? (I think it has something to do with volume but I'm unsure as your only integrating w.r.t. 2 variables so wouldn't that be like a surface area in 2 dimensions?)

Any clarification would be appreciated thanks!
Original post by amcor
So I've been learning how to do double integrals such as:

∬R (xy+4) dy dx \iint_{R}^{~} (xy+4)~dy~dx

But I'm not actually sure what this means, or represents geometrically speaking.

Anyone want to clarify what this operation is actually doing? (I think it has something to do with volume but I'm unsure as your only integrating w.r.t. 2 variables so wouldn't that be like a surface area in 2 dimensions?)

Any clarification would be appreciated thanks!


The Volume of the surface
Reply 2
Avatar for soup
soup
11
You are calculating the surface of the function xy + 4 in the region R
Reply 3
Avatar for amcor
amcor
OP
Original post by maths learner
The Volume of the surface


What is the volume of a surface though? I think my understanding is completely off because how can you have a volume in 2 dimensions isn't it just surface area?
Reply 4
Avatar for amcor
amcor
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Original post by soup
You are calculating the surface of the function xy + 4 in the region R


surface area?
Reply 5
Avatar for BabyMaths
BabyMaths
Original post by amcor
surface area?


∫abf(x)dx\int_a^b f(x) dx gives you an area between a curve and the x axis between x=a and x=b.

∫ab∫cdf(x,y)dxdy\int_a^b \int_c^d f(x,y) dx dy gives you a volume between a surface and the x-y plane in a certain region...
Reply 6
Avatar for amcor
amcor
OP
Original post by BabyMaths
∫abf(x)dx\int_a^b f(x) dx gives you an area between a curve and the x axis between x=a and x=b.

∫ab∫cdf(x,y)dxdy\int_a^b \int_c^d f(x,y) dx dy gives you a volume between a surface and the x-y plane in a certain region...



Thank you, I think that clears it up for me.
Original post by amcor
What is the volume of a surface though? I think my understanding is completely off because how can you have a volume in 2 dimensions isn't it just surface area?


Yes the surface area, sorry.
Reply 8
Avatar for steve44
steve44
9
dx dy f(x,y) is the volume of a oblong of area dx dy and height f(x,y). The integral is then the summation of these individual volumes.
(edited 10 years ago)
Reply 9
Avatar for steve44
steve44
9
It is analogous to an integral over one variable - there dx f(x) is the area of a rectangle of width dx and height f(x). The integral is then a summation over all individual areas and gives the total area under the curve.

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