If we have Cartesian coordinates, to calculate the volume in R3 we use a triple integral. But when we calculate the surface on some manifold (like the Earth surface), we need to consider some coordinates (like latitude, longtitude) that are asymptotically cartesian only in local neighborghood. This integral then has an additional multiplier (area per unit of change of parameters) that make it different from a simple double integral on a plane area.

And Vikash is also right that not all 3 dimernsional intergals have the meaning of volume. For example, we can have distribution of population w.r.t. 3 parameters: weight, height and years of education (or smth else).

Consider a quantity is product of three variables x, y, and z, and each of these variables are function of time. Then, each of the variables must be integrated over the lower and upper limits of time, using a triple integral formulation. Since the domain of integration is time, it cannot be called "volume" integral.

But if the variables were function of length, such that integration was done over length scales, then the triple integral formulation can safely be called volume integral.

Volume integration corresponds to the case when all three variables of integration in the triple integral formulation are spatial length.

To summarize: All volume integrals are triple integrals. But, all triple integrals are not volume integrals.

If we have Cartesian coordinates, to calculate the volume in R3 we use a triple integral. But when we calculate the surface on some manifold (like the Earth surface), we need to consider some coordinates (like latitude, longtitude) that are asymptotically cartesian only in local neighborghood. This integral then has an additional multiplier (area per unit of change of parameters) that make it different from a simple double integral on a plane area.

And Vikash is also right that not all 3 dimernsional intergals have the meaning of volume. For example, we can have distribution of population w.r.t. 3 parameters: weight, height and years of education (or smth else).

The concepts Simple/Single Integral, Double Integral and Triple Integral are used by default in mathematics.

Line Integral, Surface Integral and Volume Integral are used mainly in different thechnical applications fields like fluid mechanics.

Since these are algorithms that dimensionally (in an abstract geometric sense) contain the product, dx * dy * dz, they can be referred to as "volume integrals" in a general descriptive aspect.

The volume int is calculated by reducing to triple one. This action demands a system of coordinate and an order in the triple.

So, there is no one to one correspondence of the notions.

Not. The integral from the center to the surface and from the surface to the center is different.

Each volume is a triple integral or equivalent but not each triple integral is a volume.

See Tonelli Theorem.

A volume integral is originally defined in terms of limits of a sum, in much the same way as a single Riemann integral is defined, but with terms in the sums having a volume interpretation. A theorem (a separate theorem, not the definition of the volume integral, but uses that definition) then shows how the volume integral can be expressed in terms of a triple Riemann integral. I have never seen the question asked the other way (is a triple integral a volume integral?) but my hunch is that a triple integral is a volume integral in some suitably defined space. To convert a given triple integral into a volume integral in some physical kind of space, we can make that happen by using two kinds of flexibility. One is to define a transformation between Cartesian coordinates and whatever the integration variables are, and use that to calculate a Jacobian that defines a spatial volume element in some corresponding (selected by the person that invented the transformation) physical spacial region. The second flexibility allows the size and shape of this spacial region to be whatever is needed to correspond to the original three-dimensional (needed for any triple integral) integration region.

Friend ,Volume integral is a particular case of triple integral where we find volume of the given solid using triple integral.

Sita,

In physics and engineering, a volume integral is much more general than an integral that finds the volume of a solid. The integrand is often called some kind of a density (e.g., the volume integral of a mass density) but it can be just about anything. So I still think that when given some triple integral, we can invent some volume integral that is equal to it.

@Hakeem Niyas

In my view these two are different . In

some cases with constraint integrals differ. Interestingly no single book in Math discusses that .

The notion of n-volume integral has sense for any natural number n>=3.

In the particular case n=3, it coincides with that of a triple integral. In general, the former integral is more general that the latter one. The notions of n-1 surface integrals and of integral over a p-dimensional sub-variety of R^n (1

A volume integral can be written as a triple integral, where each of the three integration variables has the physical dimension of a length. Alternatively, you can regard a volume integral as an integral over a measure "dV" that has the physical dimension (length)^3. Compare for instance the cases a) dV=dx dy dz in cartesian coordinates, b) of spherical coordinates r, theta, phi where you can write dV=r^2 dr sin(theta) d theta d phi.

Obviously, such issues are related to the question how to change coordinates in multiple integrals: this question can be treated in a purely mathematical fashion without any physical considerations.

If the function is constant the second integral is area but if it another function its volume .

A triple integral is a volume integral when you integrate over spatial coordinates (X,Y, Z, or R, sigma, Z, ...) but if you are integrating for example two spatial variables (X, Y) and the time to obtain for example the energy of a system that integral even with three variables is not a volume integral.

I think the subject refers to the inegral as a mathematical entity and, in any case, geometric, but without a physical reference. Thus, these dimensions are "dimensionless" dimensions. The integral does not ask if the three dimensions have the same or different nature. For the mathematics "dx3" is consistent with "dx1" or "dx2" without inquiring about its "nature". They are only symbols.

Triple integral and volume is the same .

Basically integral is used to measure area under curve whether open or bounded. Volume integral is a particular case of Triple integral. Triple integral is used to find the volume of 3-dimensional object . To find the volume of any object we need length, breadth and height. So we need one integral to measure length, second integral for Breadth and third integral for Height. So we need triple integral to find volume. sometimes author say Volume and Triple integral is same.

But Volume can also find using double integral when we have one axis is defined or given

We often talk about an area under a curve even when the coordinates are not spatial distances. Physics will talk about the volume of a region of phase-space even though some of the coordinates are momentum coordinates. Using this generalized interpretation of a volume, but applied specifically to three dimensions, let us see if a triple integral is a volume integral. We start with the rigorous definition of a Riemann integral in three dimensions. It uses the concept of upper and lower sums and proves the convergence (with suitable restrictions on the function being integrated) of a particular process of taking a limit. It also constructs a set of parallelepipeds and uses their volumes to define the upper and lower sums. In other words, a Riemann integral in three dimensions is a volume integral in three dimensions. Theorems will show how a Riemann integral in three dimensions (i.e., a volume integral) can be expressed as a triple Riemann integral. I think (maybe I am careless here) that any triple Riemann integral is a Riemann integral in three dimensions. If so, then a triple integral and a volume integral are the same.

I should point out that a volume integral means, to me, that the definition of the integral refers to volume elements, so a volume integral (in three dimensions) is a Riemann integral in three dimensions. A volume integral is not necessarily the same as an integral that calculates a volume. A volume integral need not calculate a volume (e.g., it might be a volume integral of a mass density). An integral that calculates a volume need not be a volume integral (it might be a double integral with an integrand selected for a volume calculation).

In the Euclidean affine space, the identity between the "physical" volume and the geometric three-dimensional volume is admitted. By generalization, and without pretending an exact visualization, it does not disgust reason to denote (for conceptual purposes) as an integral of volume the triple integral in another space with generalized orthogonality (but obviusly not perpendicular).

For a triple integral to be a volume integral, there are coordinate system-dependent "scale factors" (or coordinate-dependent "scale factor functions") that need to be included within the integral. For Cartesian coordinates, the scale factor is "1" so the volume element is "dx dy dz" ; for spherical polar coordinates, the coordinate-dependent scale is "r^2 sin(theta)", with theta being the angle from the azimuth, so that the volume element is "r^2 sin(theta) dr d(phi) d(theta)". Explained similarly to this in Hildebrand, Advanced Calculus for Applications.

Jonathan has a good point. In my previous post I said " I think (maybe I am careless here) that any triple Riemann integral is a Riemann integral in three dimensions." I was careless by not recognizing the possibility of using coordinate transformations in the integrals, so the coordinates in one or more of the triple Riemann integrals need not be the same as the coordinates used in the original definition of the Riemann integrals in three dimensions. I think Jonathan fixed the mistake that I made.

Dear Hakeem Niyas , I have attached some basic diagrams and descriptions to aide in answering your question.

In a nutshell the context within which a function is integrated is critical. A function is integrated over a region (Kindly see images).

• 2D-area is f(x)*dx which can be thought of as sum of width(dx) times height(f(x)), where the width is along the x-axis.
• 3D-Volume requires a 2D function f(x,y). As such the 3D-Volume is height(f(x,y,)) * breadth(dy)*width(dx)
• 4D-'Volume' is (Hyper metric)f(x,y,z)*height(dz)* breadth(dy)*width(dx)

I think that for a complete answer of the questions to integrals the following short comparative description (not complete rigorous mathematically) of different integral types can be useful even if some things are repeated.

Simple integral, double integral and triple integral (also ordinary integrals)

All these integrals known in mathematic also as ordinary integrals can be derived easy from multiple integrals of Riemann integrable real functions of n real variables over an n dimensional integration domain D ⸦Rn (i.e. f(x1, x2, ..., xn) → R ):

∫... ∫D f (x1, x2, ..., xn) dx1...dxn

Rn is the cartesian product of n real number set R and the integral is the limit of a Riemann sum corresponding to a well-defined partition of the integration domain D ⸦Rn.

The simple integral, double integral and triple integral are cases of the above-mentioned n-dimensional multiple integral:

· Simple integral: for n=1, D ⸦R1=R, (i.e. x1=x and xi=0, i=2,3, …, n) the integration domain D is an interval of the real number set R (e.g. a ≤ x ≤b, b finite or infinite)) and the multiple integral becomes Simple integral expressed as:

∫ab f(x) dx.

Geometrically for f(x)=1 the simple integral represents the length of the considered interval (i.e. lab). On this basis the simple integral can be considered also as straight-line integral and so it is a special case of line-integral).

· Double integral: for n=2, D ⊆ R2=RxR, ( x1=x, x2=y and xi=0, i=3,4 …, n) the integration domain D is a subdomain of the cartesian product of two real number sets RxR (e.g. a ≤ x ≤b and c ≤ x ≤d)) and the multiple integral becomes double integral expressed as: ∫∫D f(x,y) dxdy.

Geometrically D is a plane surface and for f(x,y)=1 the double integral represents the area of this plan surface D (i.e. A(D)). On this basis the double integral can be considered also as plane-surface integral (i.e a special case of surface integral).

· Triple integral: for n=3, D ⊆ R3=RxRxR, (i.e. x1=x, x2=y, x3=z and

xi=0, i=4,…, n) the integration domain D is a part of the cartesian product of three real number sets RxRxR) and the multiple integral becomes triple integral expressed as: ∫∫∫D f(x,y,z)dxdydz.

Geometrically D is a volume in R3 and for f(x,y,z) =1 the triple integral represents the volume of the 3-dimensional integration domain D. The triple integral is usually applied for a direct calculation of volumes different form. On this basis the triple integral can be considered also as volume integral.

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These integrals are originally introduced for mathematical describing physical / mechanical processes of material systems. This regard continuum mechanics / fluid mechanic are representative examples whose mathematical modelling lead to differential or integral operations with scalar, vectorially or tensorial functions attached to materially systems. These functions are generally defined along of lines (curve), surfaces or volumes of general shape situated in the Euclidean space E3. The laws and principles of system-processes expressed in global form which the mathematical models of system represent lead to line integrals, surface integral and volume integrals.

The formal definition of these integrals closely follows that of ordinary integrals and can be considered as the limit of Riemann sums, but expressed directly on spatial curves, surfaces and volumes.

· Line or path integral is an integral which a function of scalar, vector or tensor nature related to space-domain of the researched material system along a space-curve C between two given points A and B integrates. Usually notations for a scalar function Φ and a vectorially function F defined along the curve C:

(e.g. ∫C Φdl; ∫C F∙dl; ∫C Fxdl )

If A and B coincide the curve C forms a loop.

The line integral can be evaluated by expressing the curve C in parametric form which in terms of a parameter is equivalent to stretching the curve C into a straight line (i.e. simple integral). If the integrand Φ=1 the line integral represents the length of curve (e.g. lAB).

· Surface integral is an integral which a function of scalar, vector or tensor nature related to space-domain of the researched material system along a space-surface S integrates. Usually notations for a scalar function Φ and a vectorially function F defined on the surface S:

(e.g. ∫S Φds; ∫S F∙ds; ∫S Fxds )

All the above integrals are taken over some surface S, which may be either open or closed.

To evaluate surface integrals over a general surface S it is usual to considering the projections of the surface onto the coordinate planes in Cartesian coordinates. If the integrand Φ=1 the surface integral represents the area of the surface A(S).

There are many examples of surface integrals in the physical sciences. Surface integrals involving vectors (the second form above) can represent a flux through S such as mass of fluid crossing the surface, electromagnetic flux of energy out of a given volume V bounded by a surface S and so on.

In the case of a plan surface S the surface integral becomes an ordinary double integral.

· Volume Integral: A Volume integral is an integral where a given function (scalar or vectorially) is integrated directly (e.g. as limit of Riemann sum for a arbitrary partition of the volume V) without use ta coordinate system, expressed as: ∫V Φ(r)dv; ∫V F(r)dV

Examples for applications of the volume integral in this general form: for Φ=ρ(r) we obtain the mass of fluid occupying the volume V; for F(r)= ρ(r )v we obtain the total linear momentum of that same fluid, where v(r) is the velocity field in the fluid and ρ(r) is the fluid density.

Relating these generalised form of space integrals there are several integral theorems like the divergence integral theorem relates the total flux of a vector field out of a closed surface S to the integral of the divergence of the vector field over the enclosed volume V.

At the same time on observe that in cartesian coordinate system the first volume integral can be expressed as ordinary triple integral. The second volume integral can be also expressed as sum of three triple integral. So, we can conclude that in opposite to line integral and surface integral the volume integral is nearly to the ordinary triple integral.

Not. In the triple integral, the coordinates are not connected, but are connected in volume.

The discussion illustrates that the question is not that trivial as it seems at the beginning. The modern conception of the integral was given by @ de Rham in fiftees. This conception distiguishes integrals of odd differential forms on smooth manifolds (like weight densities, charge densities) and that of even differential forms on oriented manifolds (like current of a flow through a surface). The advantage of the first class is that it is defined on arbitrary (orieted or not) manifold. The advatage of the second class is that the integrals obey theorems of Newton-Leibniz, Cauchy, Gauss, Stokes etc. which provide important tools for calculations of integrals of the first class.

Definitions of both types of integral are given by partition of unity, localization and reduction to multiple integrations which is of course a common wisdom.

If you have Henstock integral then the volume integral implies the triple integral but not conversely. For the Lebesgue integral see Tonelli theorem.