What Do Different Types of Integrals Represent in Calculus?

[uzman1243]

I'm doing calclus 3 right now and I'm trying to put together the results of integrals.
Can you correct me if I'm wrong and the one's I missed ( particularly 4 / 5 / 6). I also that the integrals can mean different things based on context. But in terms of areas and volumes atleast?

1) ∫ dx {limits x=a to x=b} -> gives length of a line segment

2) ∫ f(x) dx {limits x=a to x=b} -> area of region below y= f(x) >= 0 and y =0

3) ∫ ∫ dy dx {limits x=a to x=b / y= f₁ x and y = f₂x} -> area between 2 curves

4) ∫ ∫ f(x,y) = z dy dx {limits x=a to x=b / y= f₁ x and y = f₂x} -> volume bounded by the surface

5) ∫ ∫ ∫ dx dy dz -> ?

6) ∫ ∫ ∫ f(x,y,z) dx dy dz -> ?

 

[HallsofIvy]

Those are possible applications of the different kinds of integrals but you should understand that the integrals themselves do not automatically give any specific application.

) ∫ dx {limits x=a to x=b} -> gives length of a line segment.
This is equal to b- a which would be the length on the x-axis from a to b

2) ∫ f(x) dx {limits x=a to x=b} -> area of region below y= f(x) >= 0 and y =0
Assuming that y= f(x) is a graph such that f(x)> 0 for all x between a and b, then, yes, it is the area bounded by the graphs of y= f(x), y= 0, x= a, and x= b.

3) ∫ ∫ dy dx {limits x=a to x=b / y= f1 x and y = f2x} -> area between 2 curves
Yes, if y= f1(x) and y= f(x2) are such that f2(x)> f1(x) (or vice-versa) for all x between a and b, this is one possible interpretation.

4) ∫ ∫ f(x,y) = z dy dx {limits x=a to x=b / y= f1 x and y = f2x} -> volume bounded by the surfaceY
Yes, again with the stipulation that f(x,y) > 0 for all x, y in that area. Of course, you could also interpret f(x,y) as "surface density" so that this integral is the "mass" of the surface. More generally, f(x,y) could be a "density" of any property and the integral the total property. For example f could be temperature which is proportional to heat density and then the integral would be the heat contained in the surface.

5) ∫ ∫ ∫ dx dy dz -> ?[/quote]
The simplest interpretation would be the volume of the region integrated over.

6) ∫ ∫ ∫ f(x,y,z) dx dy dz -> ?
We can interpret f(x, y, z) as the density of some property - mass density, temperature as a "heat density", etc so that the integral gives the "total" mass or heat, etc.

 

[uzman1243]

Those are possible applications of the different kinds of integrals but you should understand that the integrals themselves do not automatically give any specific application.

) ∫ dx {limits x=a to x=b} -> gives length of a line segment.
This is equal to b- a which would be the length on the x-axis from a to b

2) ∫ f(x) dx {limits x=a to x=b} -> area of region below y= f(x) >= 0 and y =0
Assuming that y= f(x) is a graph such that f(x)> 0 for all x between a and b, then, yes, it is the area bounded by the graphs of y= f(x), y= 0, x= a, and x= b.

3) ∫ ∫ dy dx {limits x=a to x=b / y= f1 x and y = f2x} -> area between 2 curves
Yes, if y= f1(x) and y= f(x2) are such that f2(x)> f1(x) (or vice-versa) for all x between a and b, this is one possible interpretation.

4) ∫ ∫ f(x,y) = z dy dx {limits x=a to x=b / y= f1 x and y = f2x} -> volume bounded by the surfaceY
Yes, again with the stipulation that f(x,y) > 0 for all x, y in that area. Of course, you could also interpret f(x,y) as "surface density" so that this integral is the "mass" of the surface. More generally, f(x,y) could be a "density" of any property and the integral the total property. For example f could be temperature which is proportional to heat density and then the integral would be the heat contained in the surface.

5) ∫ ∫ ∫ dx dy dz -> ?
The simplest interpretation would be the volume of the region integrated over.

6) ∫ ∫ ∫ f(x,y,z) dx dy dz -> ?
We can interpret f(x, y, z) as the density of some property - mass density, temperature as a "heat density", etc so that the integral gives the "total" mass or heat, etc.

Thank you so much!