# What is double integral by interpretation?

1. Aug 6, 2015

### 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!

2. Aug 6, 2015

### Hesch

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 ).

3. Aug 6, 2015

### vjacheslav

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

4. Aug 6, 2015

### Hesch

No, the integrals of second and third order are separated examples ( stand alones ).

5. Aug 6, 2015

### vjacheslav

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

6. Aug 6, 2015

### 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 ).

7. Aug 6, 2015

### 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

8. Aug 6, 2015

### Staff: Mentor

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.

9. Aug 6, 2015

### 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?

10. Aug 6, 2015

### Staff: Mentor

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.

11. Aug 7, 2015

### 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?

12. Aug 7, 2015

### Staff: Mentor

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.

13. Aug 8, 2015

### 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!

14. Aug 8, 2015

### Hesch

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?
Thus the known formula: s = ½at2

Last edited: Aug 8, 2015