Double integral as volume under a surface z = 10 − x2 − y2/8.
The rectangular region at the bottom of the body is the domain of
integration, while the surface is the graph of the two-variable function
to be integrated.
The multiple integral is a definite integral of a function of more than one real variable, for example, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in R2 are called double integrals, and integrals of a function of three variables over a region of R3 are called triple integrals.
Introduction
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where z = f(x, y)) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions.
Multiple integration of a function in n variables: f(x1, x2, ..., xn) over a domain D
is most commonly represented by nested integral signs in the reverse
order of execution (the leftmost integral sign is computed last),
followed by the function and integrand arguments in proper order (the
integral with respect to the rightmost argument is computed last). The
domain of integration is either represented symbolically for every
argument over each integral sign, or is abbreviated by a variable at the
rightmost integral sign:
Since the concept of an antiderivative is only defined for functions of a single real variable, the usual definition of the indefinite integral does not immediately extend to the multiple integral.
Mathematical definition
For n > 1, consider a so-called "half-open" n-dimensional hyperrectangular domain T, defined as:
Partition each interval [aj, bj) into a finite family Ij of non-overlapping subintervals ijα, with each subinterval closed at the left end, and open at the right end.
Then the finite family of subrectangles C given by
is a partition of T; that is, the subrectangles Ck are non-overlapping and their union is T.
Let f : T → R be a function defined on T. Consider a partition C of T as defined above, such that C is a family of m subrectangles Cm and
We can approximate the total (n + 1)th-dimensional volume bounded below by the n-dimensional hyperrectangle T and above by the n-dimensional graph of f with the following Riemann sum:
where Pk is a point in Ck and m(Ck) is the product of the lengths of the intervals whose Cartesian product is Ck, also known as the measure of Ck.
The diameter of a subrectangle Ck is the largest of the lengths of the intervals whose Cartesian product is Ck. The diameter of a given partition of T
is defined as the largest of the diameters of the subrectangles in the
partition. Intuitively, as the diameter of the partition C is restricted smaller and smaller, the number of subrectangles m gets larger, and the measure m(Ck) of each subrectangle grows smaller. The function f is said to be Riemann integrable if the limit
exists, where the limit is taken over all possible partitions of T of diameter at most δ.
If f is Riemann integrable, S is called the Riemann integral of f over T and is denoted
Frequently this notation is abbreviated as
where x represents the n-tuple (x1, ... xn) and dnx is the n-dimensional volume differential.
The Riemann integral of a function defined over an arbitrary bounded n-dimensional
set can be defined by extending that function to a function defined
over a half-open rectangle whose values are zero outside the domain of
the original function. Then the integral of the original function over
the original domain is defined to be the integral of the extended
function over its rectangular domain, if it exists.
In what follows the Riemann integral in n dimensions will be called the multiple integral.
Properties
Multiple
integrals have many properties common to those of integrals of
functions of one variable (linearity, commutativity, monotonicity, and
so on). One important property of multiple integrals is that the value
of an integral is independent of the order of integrands under certain
conditions. This property is popularly known as Fubini's theorem.
Particular cases
In the case of T ⊆ R2, the integral
is the double integral of f on T, and if T ⊆ R3 the integral
is the triple integral of f on T.
Notice that, by convention, the double integral has two integral
signs, and the triple integral has three; this is a notational
convention which is convenient when computing a multiple integral as an
iterated integral, as shown later in this article.
Methods of integration
The
resolution of problems with multiple integrals consists, in most cases,
of finding a way to reduce the multiple integral to an iterated integral, a series of integrals of one variable, each being directly solvable. For continuous functions, this is justified by Fubini's theorem. Sometimes, it is possible to obtain the result of the integration by direct examination without any calculations.
The following are some simple methods of integration.
Integrating constant functions
When the integrand is a constant functionc, the integral is equal to the product of c and the measure of the domain of integration. If c = 1 and the domain is a subregion of R2, the integral gives the area of the region, while if the domain is a subregion of R3, the integral gives the volume of the region.
Example. Let f(x, y) = 2 and
in which case
,
since by definition we have:
Use of symmetry
When
the domain of integration is symmetric about the origin with respect to
at least one of the variables of integration and the integrand is odd
with respect to this variable, the integral is equal to zero, as the
integrals over the two halves of the domain have the same absolute value
but opposite signs. When the integrand is even
with respect to this variable, the integral is equal to twice the
integral over one half of the domain, as the integrals over the two
halves of the domain are equal.
Example 1. Consider the function f(x,y) = 2 sin(x) − 3y3 + 5 integrated over the domain
a disc with radius 1 centered at the origin with the boundary included.
Using the linearity property, the integral can be decomposed into three pieces:
The function 2 sin(x) is an odd function in the variable x and the disc T is symmetric with respect to the y-axis, so the value of the first integral is 0. Similarly, the function 3y3 is an odd function of y, and T is symmetric with respect to the x-axis,
and so the only contribution to the final result is that of the third
integral. Therefore the original integral is equal to the area of the
disk times 5, or 5π.
Example 2. Consider the function f(x, y, z) = x exp(y2 + z2) and as integration region the sphere with radius 2 centered at the origin,
The "ball" is symmetric about all three axes, but it is sufficient to integrate with respect to x-axis to show that the integral is 0, because the function is an odd function of that variable.
Normal domains on R2
This method is applicable to any domain D for which:
the projection of D onto either the x-axis or the y-axis is bounded by the two values, a and b
any line perpendicular to this axis that passes between these two
values intersects the domain in an interval whose endpoints are given by
the graphs of two functions, α and β.
Such a domain will be here called a normal domain. Elsewhere
in the literature, normal domains are sometimes called type I or type II
domains, depending on which axis the domain is fibred over. In all
cases, the function to be integrated must be Riemann integrable on the
domain, which is true (for instance) if the function is continuous.
x-axis
If the domain D is normal with respect to the x-axis, and f : D → R is a continuous function; then α(x) and β(x) (both of which are defined on the interval [a, b]) are the two functions that determine D. Then, by Fubini's theorem:
y-axis
If D is normal with respect to the y-axis and f : D → R is a continuous function; then α(y) and β(y) (both of which are defined on the interval [a, b]) are the two functions that determine D. Again, by Fubini's theorem:
Normal domains on R3
If T is a domain that is normal with respect to the xy-plane and determined by the functions α(x, y) and β(x, y), then
This definition is the same for the other five normality cases on R3. It can be generalized in a straightforward way to domains in Rn.
Change of variables
The limits of integration are often not easily interchangeable
(without normality or with complex formulae to integrate). One makes a change of variables
to rewrite the integral in a more "comfortable" region, which can be
described in simpler formulae. To do so, the function must be adapted to
the new coordinates.
Example 1a. The function is f(x, y) = (x − 1)2 + √y; if one adopts the substitution x′ = x − 1, y′ = y therefore x = x′ + 1, y = y′ one obtains the new function f2(x, y) = (x′)2 + √y.
Similarly for the domain because it is delimited by the original variables that were transformed before (x and y in example).
the differentials dx and dy transform via the absolute value of the determinant of the Jacobian matrix
containing the partial derivatives of the transformations regarding the
new variable (consider, as an example, the differential transformation
in polar coordinates).
There exist three main "kinds" of changes of variable (one in R2, two in R3); however, more general substitutions can be made using the same principle.
Polar coordinates
Transformation from cartesian to polar coordinates.
In R2 if the domain has a circular symmetry and the function has some particular characteristics one can apply the transformation to polar coordinates (see the example in the picture) which means that the generic points P(x, y)
in Cartesian coordinates switch to their respective points in polar
coordinates. That allows one to change the shape of the domain and
simplify the operations.
The fundamental relation to make the transformation is the following:
Example 2a. The function is f(x, y) = x + y and applying the transformation one obtains
Example 2b. The function is f(x, y) = x2 + y2, in this case one has:
The transformation of the domain is made by defining the radius'
crown length and the amplitude of the described angle to define the ρ, φ intervals starting from x, y.
Example of a domain transformation from cartesian to polar.
Example 2c. The domain is D = {x2 + y2 ≤ 4}, that is a circumference of radius 2; it's evident that the covered angle is the circle angle, so φ varies from 0 to 2π, while the crown radius varies from 0 to 2 (the crown with the inside radius null is just a circle).
Example 2d. The domain is D = {x2 + y2 ≤ 9, x2 + y2 ≥ 4, y ≥ 0}, that is the circular crown in the positive y half-plane (please see the picture in the example); φ describes a plane angle while ρ varies from 2 to 3. Therefore the transformed domain will be the following rectangle:
which has been obtained by inserting the partial derivatives of x = ρ cos(φ), y = ρ sin(φ) in the first column respect to ρ and in the second respect to φ, so the dx dy differentials in this transformation become ρ dρ dφ.
Once the function is transformed and the domain evaluated, it is
possible to define the formula for the change of variables in polar
coordinates:
φ is valid in the [0, 2π] interval while ρ, which is a measure of a length, can only have positive values.
Example 2e. The function is f(x, y) = x and the domain is the same as in Example 2d. From the previous analysis of D we know the intervals of ρ (from 2 to 3) and of φ (from 0 to π). Now we change the function:
finally let's apply the integration formula:
Once the intervals are known, you have
Cylindrical coordinates
Cylindrical coordinates.
In R3 the integration on domains with a circular base can be made by the passage to cylindrical coordinates; the transformation of the function is made by the following relation:
The domain transformation can be graphically attained, because only
the shape of the base varies, while the height follows the shape of the
starting region.
Example 3a. The region is D = {x2 + y2 ≤ 9, x2 + y2 ≥ 4, 0 ≤ z ≤ 5}
(that is the "tube" whose base is the circular crown of Example 2d and
whose height is 5); if the transformation is applied, this region is
obtained:
(that is, the parallelepiped whose base is similar to the rectangle in Example 2d and whose height is 5).
Because the z component is unvaried during the transformation, the dx dy dz differentials vary as in the passage to polar coordinates: therefore, they become ρ dρ dφ dz.
Finally, it is possible to apply the final formula to cylindrical coordinates:
This method is convenient in case of cylindrical or conical domains or in regions where it is easy to individuate the z interval and even transform the circular base and the function.
Example 3b. The function is f(x, y, z) = x2 + y2 + z and as integration domain this cylinder: D = {x2 + y2 ≤ 9, −5 ≤ z ≤ 5 }. The transformation of D in cylindrical coordinates is the following:
while the function becomes
Finally one can apply the integration formula:
developing the formula you have
Spherical coordinates
Spherical coordinates.
In R3 some domains
have a spherical symmetry, so it's possible to specify the coordinates
of every point of the integration region by two angles and one distance.
It's possible to use therefore the passage to spherical coordinates; the function is transformed by this relation:
Points on the z-axis do not have a precise characterization in spherical coordinates, so θ can vary between 0 and 2π.
The better integration domain for this passage is obviously the sphere.
Example 4a. The domain is D = x2 + y2 + z2 ≤ 16 (sphere with radius 4 and center at the origin); applying the transformation you get the region
The Jacobian determinant of this transformation is the following:
The dx dy dz differentials therefore are transformed to ρ2 sin(φ) dρdθdφ.
This yields the final integration formula:
It is better to use this method in case of spherical domains and in case of functions that can be easily simplified by the first fundamental relation of trigonometry extended to R3 (see Example 4b); in other cases it can be better to use cylindrical coordinates (see Example 4c).
The extra ρ2 and sin φ come from the Jacobian.
In the following examples the roles of φ and θ have been reversed.
Example 4b.D is the same region as in Example 4a and f(x, y, z) = x2 + y2 + z2 is the function to integrate. Its transformation is very easy:
while we know the intervals of the transformed region T from D:
We therefore apply the integration formula:
and, developing, we get
Example 4c. The domain D is the ball with center at the origin and radius 3a,
and f(x, y, z) = x2 + y2 is the function to integrate.
Looking at the domain, it seems convenient to adopt the passage
to spherical coordinates, in fact, the intervals of the variables that
delimit the new T region are obviously:
However, applying the transformation, we get
.
Applying the formula for integration we obtain:
which is very hard to solve. This problem will be solved by using the passage to cylindrical coordinates. The new T intervals are
the z interval has been obtained by dividing the ball into two hemispheres simply by solving the inequality from the formula of D (and then directly transforming x2 + y2 into ρ2). The new function is simply ρ2. Applying the integration formula
.
Then we get
Thanks to the passage to cylindrical coordinates it was possible to
reduce the triple integral to an easier one-variable integral.
Examples
Double integral over a rectangle
Let us assume that we wish to integrate a multivariable function f over a region A:
From this we formulate the iterated integral
The inner integral is performed first, integrating with respect to x and taking y as a constant, as it is not the variable of integration. The result of this integral, which is a function depending only on y, is then integrated with respect to y.
We then integrate the result with respect to y.
Sometimes, the order of integration is interchangeable, that is, integrating with respect to x first and integrating with respect to y first produce the same result. For example, doing the previous calculation with order reversed gives the same result:
The instances where the order is interchangeable is determined by Fubini's theorem.
Double integral over a normal domain
Example: double integral over the normal region D
Consider the region (please see the graphic in the example):
Calculate
This domain is normal with respect to both the x- and y-axes. To apply the formulae it is required to find the functions that determine D and the intervals over which these functions are defined. In this case the two functions are:
while the interval is given by the intersections of the functions with x = 0, so the interval is [a, b] = [0, 1] (normality has been chosen with respect to the x-axis for a better visual understanding).
It is now possible to apply the formula:
(at first the second integral is calculated considering x as a constant). The remaining operations consist of applying the basic techniques of integration:
If we choose normality with respect to the y-axis we could calculate
and obtain the same value.
Example of domain in R3 that is normal with respect to the xy-plane.
Calculating volume
Using the methods previously described, it is possible to calculate the volumes of some common solids.
Cylinder: The volume of a cylinder with height h and circular base of radius R can be calculated by integrating the constant function h over the circular base, using polar coordinates.
This is in agreement with the formula for the volume of a prism
.
Sphere: The volume of a sphere with radius R can be calculated by integrating the constant function 1 over the sphere, using spherical coordinates.
Tetrahedron (triangular pyramid or 3-simplex): The volume of a tetrahedron with its apex at the origin and edges of length l along the x-, y- and z-axes can be calculated by integrating the constant function 1 over the tetrahedron.
This is in agreement with the formula for the volume of a pyramid
Example of an improper domain.
Multiple improper integral
In case of unbounded domains or functions not bounded near the boundary of the domain, we have to introduce the double improper integral or the triple improper integral.
If the integral is not absolutely convergent, care is needed not to confuse the concepts of multiple integral and iterated integral, especially since the same notation is often used for either concept. The notation
means, in some cases, an iterated integral rather than a true double integral. In an iterated integral, the outer integral
is the integral with respect to x of the following function of x:
A double integral, on the other hand, is defined with respect to area in the xy-plane. If the double integral exists, then it is equal to each of the two iterated integrals (either "dy dx" or "dx dy")
and one often computes it by computing either of the iterated
integrals. But sometimes the two iterated integrals exist when the
double integral does not, and in some such cases the two iterated
integrals are different numbers, i.e., one has
On the other hand, some conditions ensure that the two iterated
integrals are equal even though the double integral need not exist. By
the Fichtenholz–Lichtenstein theorem, if f is bounded on [0, 1] × [0, 1]
and both iterated integrals exist, then they are equal. Moreover,
existence of the inner integrals ensures existence of the outer
integrals. The double integral need not exist in this case even as Lebesgue integral, according to Sierpiński.
The notation
may be used if one wishes to be emphatic about intending a double integral rather than an iterated integral.
Some practical applications
Quite
generally, just as in one variable, one can use the multiple integral
to find the average of a function over a given set. Given a set D ⊆ Rn and an integrable function f over D, the average value of f over its domain is given by
Additionally, multiple integrals are used in many applications in physics. The examples below also show some variations in the notation.
In mechanics, the moment of inertia is calculated as the volume integral (triple integral) of the density weighed with the square of the distance from the axis:
If there is a continuous function ρ(x) representing the density of the distribution at x, so that dm(x) = ρ(x)d3x, where d3x is the Euclidean volume element, then the gravitational potential is
In electromagnetism, Maxwell's equations can be written using multiple integrals to calculate the total magnetic and electric fields.[11] In the following example, the electric field produced by a distribution of charges given by the volume charge densityρ( r→ ) is obtained by a triple integral of a vector function:
This can also be written as an integral with respect to a signed measure representing the charge distribution.
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container; i. e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.
Three dimensional
mathematical shapes are also assigned volumes. Volumes of some simple
shapes, such as regular, straight-edged, and circular shapes can be
easily calculated using arithmeticformulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement.
Displacement of liquid can also be used to determine the volume of a
gas. The combined volume of two substances is usually greater than the
volume of just one of the substances. However, sometimes one substance
dissolves in the other and in such cases the combined volume is not additive.
Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre (cm3) is the volume of a cube whose sides are one centimetre (1 cm) in length.
In the International System of Units (SI), the standard unit of volume is the cubic metre (m3). The metric system also includes the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus
Capacity is defined by the Oxford English Dictionary
as "the measure applied to the content of a vessel, and to liquids,
grain, or the like, which take the shape of that which holds them". (The word capacity has other unrelated meanings, as in e.g. capacity management.)
Capacity is not identical in meaning to volume, though closely related;
the capacity of a container is always the volume in its interior. Units
of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length.
In SI the units of volume and capacity are closely related: one litre
is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side.
In other systems the conversion is not trivial; the capacity of a
vehicle's fuel tank is rarely stated in cubic feet, for example, but in
gallons (an imperial gallon fills a volume of 0.1605 cu ft).
The density of an object is defined as the ratio of the mass to the volume. The inverse of density is specific volume which is defined as volume divided by mass. Specific volume is a concept important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied.
The volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second [m3 s−1]).
and the volume integral in spherical coordinates (using the convention for angles with as the azimuth and measured from the polar axis; see more on conventions) has the form
h = any dimension of the figure, A(h) = area of the cross-sections perpendicular to h described as a function of the position along h. a and b are the limits of integration for the volumetric sweep. (This will work for any figure if its cross-sectional area can be determined from h).
Any rotated figure (washer method; calculus required)
and are functions expressing the outer and inner radii of the function, respectively.
Volume ratios for a cone, sphere and cylinder of the same radius and height
A cone, sphere and cylinder of radius r and height h
The above formulas can be used to show that the volumes of a cone, sphere and cylinder of the same radius and height are in the ratio 1 : 2 : 3, as follows.
Let the radius be r and the height be h (which is 2r for the sphere), then the volume of cone is
the volume of the sphere is
while the volume of the cylinder is
The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes.
Volume formula derivations
Sphere
The volume of a sphere is the integral of an infinite number of infinitesimally small circular disks of thickness dx. The calculation for the volume of a sphere with center 0 and radius r is as follows.
The surface area of the circular disk is .
The radius of the circular disks, defined such that the x-axis cuts perpendicularly through them, is
or
where y or z can be taken to represent the radius of a disk at a particular x value.
Using y as the disk radius, the volume of the sphere can be calculated as
Now
Combining yields
This formula can be derived more quickly using the formula for the sphere's surface area, which is . The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to
Cone
The cone is a
type of pyramidal shape. The fundamental equation for pyramids,
one-third times base times altitude, applies to cones as well.
However, using calculus, the volume of a cone is the integral of an infinite number of infinitesimally thin circular disks of thickness dx. The calculation for the volume of a cone of height h, whose base is centered at (0, 0, 0) with radius r, is as follows.
The radius of each circular disk is r if x = 0 and 0 if x = h, and varying linearly in between—that is,
The surface area of the circular disk is then
The volume of the cone can then be calculated as
and after extraction of the constants
Integrating gives us
Polyhedron
Volume in differential geometry
In differential geometry, a branch of mathematics, a volume form on a differentiable manifold is a differential form
of top degree (i.e., whose degree is equal to the dimension of the
manifold) that is nowhere equal to zero. A manifold has a volume form
if and only if it is orientable. An orientable manifold has infinitely
many volume forms, since multiplying a volume form by a non-vanishing
function yields another volume form. On non-orientable manifolds, one
may instead define the weaker notion of a density. Integrating the volume form gives the volume of the manifold according to that form.
where the are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold, and is the determinant of the matrix representation of the metric tensor on the manifold in terms of the same basis.