What is double integral by interpretation?

[vjacheslav]

Very simple question for you, friends.
As is well known, usual integral has interpretation as square under function's graphic.
Then, what is double (and triple) integral by analogue?
Thanks!

 

[Hesch]

Then, what is double (and triple) integral by analogue?

Say

f(x) = k. ( a constant)

g(x) = ∫ f(x) dx = k*x. ( a ramp ).

h(x) = ∫ g(x) dx = ½*k*x². ( a polynomial ).

So ∫∫ f(x) dx² is the area under the ramp.

Within physics you could say ( a = acceleration , v = velocity , p = position ):

v(t) = ∫ a(t) dt

p(t) = ∫ v(t) dt

So p(t) = ∫∫ a(t) dt²

Often triple integrals are used to calculate volume, e.g. the volume of a ball = 4/3 * π * r³ ( 3 dimensions ).

 

[vjacheslav]

Thanks, Hesch!
But seems like you interchange the double and triple integral by second and third polynomial power...

 

[Hesch]

But seems like you interchange the double and triple integral by second and third polynomial power...

No, the integrals of second and third order are separated examples ( stand alones ).
Just ask again if I have misunderstood your comment.

 

[vjacheslav]

Some misunderstood arised, as I see. Double = dxdy triple = dxdydz ,
no second and third order, sorry.

 

[Hesch]

You are absolutely right. I should have pointed that out.

And speaking of a volume of a ball, it could be calculated by a double integral, but I hope you understand the idea in calculating volumes generally.
( 3 dimensions: dxdydz ).

 

[vjacheslav]

So Int(f(x)dx) = 2 dim
Accordingly to you
Int(f(x,y,z)dxdydz) = 3 dim
and the very question is
Int(f(x,y)dxdy) = ? dim

 

[Mark44]

So Int(f(x)dx) = 2 dim
Accordingly to you
Int(f(x,y,z)dxdydz) = 3 dim
and the very question is
Int(f(x,y)dxdy) = ? dim

It depends on the integrand. A triple integral with an integrand of 1 would give the volume (three dimensions) of the region over which integration is performed. A double integral could also give the volume of some region if the limits of integration represented an area and the integrand represented the height of the region.

Iterated integrals (either double or triple) don't necessarily have to represent area and volume, respectively. They could represent the mass of some three-dimensional solid, as well as many other possible applications of these integrals.

 

[vjacheslav]

Grateful for you answer, but it still remains discussible. For ex, integrand 1 (f(x) as I see?) taken on dxdydz. How many dims it will give in answer? f(x)dx = 2 dim and f(x)dxdydz = 4 dim.
Am I mistaken?

 

[Mark44]

Grateful for you answer, but it still remains discussible. For ex, integrand 1 (f(x) as I see?) taken on dxdydz. How many dims it will give in answer? f(x)dx = 2 dim and f(x)dxdydz = 4 dim.
Am I mistaken?

Yes, in many cases. dx, dy, and dz typically represent length dimensions, but the integrand function does not have to have a length dimension associated with it. As I mentioned before, an integral could represent something other than length, area, or volume. For instance, it could represent the amount of work done, the mass of a three dimensional region, as well as many other possible interpretations.

 

[vjacheslav]

Whatsoever single integral represent, Int(f(x)dx) = Square under function's graphic, isn't it? Could double or triple integral represent such an analogy?

 

[Mark44]

Whatsoever single integral represent, Int(f(x)dx) = Square under function's graphic, isn't it?

Again, not necessarily. An integral ##\int_a^b f(x) dx## is a number. It doesn't have to represent area. As I said before, it could represent the amount of work done in moving something from x = a to x = b, or it could represent the average value of a function (if in the form ##\frac 1 {b - a} \int_a^b f(x) dx##. It could represent volume if f(x) is the cross-sectional area. It could represent the total charge along a conductor if f(x) represents the charge density per unit length.

The bottom line is that an integral such as this does not necessarily represent area.

Could double or triple integral represent such an analogy?

 

[vjacheslav]

So nothing could be found by analogy? Pity, but I will try still in nearest future.
And now let's close the theme.
Thanks to everybody committed!

 

[Hesch]

Some misunderstood arised, as I see. Double = dxdy triple = dxdydz

A double intgral could be dxdx = dx² and a triple integral could be dxdxdx = dx³.

So I don't see why #2 isn't an analogy?

So p(t) = ∫∫ a(t) dt²

Thus the known formula: s = ½at²