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Tips and Techniques: Multiple Integrals in MapleRobert J. Lopez
Emeritus Professor of Mathematics and Maple Fellow\251 Maplesoft, a division of Waterloo Maple Inc., 2005<Text-field layout="Heading 1" style="Heading 1">Introduction</Text-field>At top level, Maple implements iterated integration by composing the Int or int commands. The older student package provided the Doubleint and Tripleint commands for iterated double and triple integrals, respectively. The MultivariateCalculus subpackage of the newer Student package provides a MultiInt command for writing iterated integrals, while the VectorCalculus package modifies the int command to accomplish the same thing. In addition, the VectorCalculus package contains the SurfaceInt and Flux commands, both capable of writing specialized iterated integrals.In this article, we will contrast the syntax and applicability of the various commands Maple has for writing and evaluating iterated integrals.<Text-field layout="Heading 1" style="Heading 1">Top Level</Text-field>At top level, the inert iterated integral NiMtSSRJbnRHNiI2JC1GJDYkLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YtOy0mRi02IyIiIjYjRiwtJkYtNiMiIiNGNC9GLDsmRixGMiZGLEY3is formed with the syntaxInt(Int(f(x,y), y=y[1](x)..y[2](x)), x=x[1]..x[2]);and evaluated with the value command. For example, to integratef := x*y;over NiNJIlJHNiI=, the first-quadrant region bounded by the curves y1 := x^2;
y2 := x;we could writeV := Int(Int(f, y=y1..y2), x=0..1);and evaluate the itegral withvalue(V);<Text-field layout="Heading 1" style="Heading 1"><Font italic="true">Doubleint</Font> and <Font italic="true">Tripleint</Font> from the <Font italic="true">student</Font> Package</Text-field>Many long-time users of Maple are probably very familiar with the Doubleint and Tripleint commands from the student package.Despite their names, these commands implemented iterated integrals, not multiple integrals. (Multiple integrals exist that cannot be evaluated by iteration, and likewise, existence of the iterated integrals does not guarantee the multiple integral exists.)Thus, the Doubleint command could be used to write the iterated integral student[Doubleint](f, y=y1..y2, x=0..1);Note that the ordering of the variables in Doubleint and Tripleint is consistent with that used when composing the Int command. It is the order in which one thinks out the bounds for the iterated integrals. The inner integral must have its bounds determined first, and that is the integration range that is supplied to either command first. The second set of bounds that must be thought out belong to the outer integral, and that range is entered second. I always considered this the "natural" order for writing the syntax of an iterated integral. That the outer integral is to the left of the inner integral, as with all operator notation, makes for an inherent reversal of the ranges, just as the operator notation for partial derivatives reverses the order of the variables used in subscript notation. Thus, just as students must understand the notationNiMvLUklZGlmZkdJKnByb3RlY3RlZEdGJjYlLUkiZkc2IjYkSSJ4R0YqSSJ5R0YqRixGLSZGKTYjSSN4eUdGKg==they likewise must understand that the order in which the ranges in an iterated integral are thought out is the opposite of the order in which they appear. However, the inner integral is both thought out and executed first, with the outer integral second in both regards.Finally, tecall that the Doubleint and Tripleint commands always wrote the inert form of the iterated integral. Compared to the top-level usage, these commands not only saved writing a parenthesis or two, but also could be used for writing notation of the formstudent[Doubleint](g(x,y),y,x,R);<Text-field layout="Heading 1" style="Heading 1">The <Font italic="true">MultiInt</Font> Command</Text-field>An iterated integral can be written with the MultiInt command from the MultivariateCalculus subpackage of the Student package. This subpackage is made accessible withrestart;
with(Student[MultivariateCalculus]):To write the iterated integralNiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUYkNiQtRiQ2JC1JImZHRig2JUkieEdGKEkieUdGKEkiekdGKC9GMzstJkYzNiMiIiI2JEYxRjItJkYzNiMiIiNGOi9GMjstJkYyRjg2I0YxLSZGMkY9RkMvRjE7JkYxRjgmRjFGPQ==using the MultiInt command, the user must know that the order in which the ranges of integration are entered in the "natural" order described above. Thus, to integrate f := x*y*z;over V, the cylinder whose projection in the xy-plane is the region bounded byy1 := x^2;
y2 := x;and which is bounded below and above by the surfacesz1 := -4-x-y;
z2 := 4-x^2-y^2;use the syntaxMultiInt(f, z=z1..z2, y=y1..y2, x=0..1, output=integral);Notice that to obtain the intert integral, the special output parameter "integral" must be included. Without this parameter, the integration is immediate, as if the integral had been written with a composition of int commands.MultiInt(f, z=z1..z2, y=y1..y2, x=0..1);Writing notation such asNiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUYkNiQtSSJnR0YoNiRJInhHRihJInlHRigvRjA7SSJSR0YoSSFHRigvRi87RjRGNA==requires the more complex syntaxMultiInt(g(x,y), y=``..R, x=``..` `);where the space between the back quotes in the second range is essential. However, this inconvenience is counterbalanced by an increased functionality for handling polar, cylindrical, and spherical coordinates.Thus, to integratef := r*cos(3*theta)^2;over the unit circle centered at the origin, one need only writeMultiInt(f, r=0..1, theta=0..2*Pi, coordinates=polar[r,theta]);The inert form of this integral appears asq := MultiInt(f, r=0..1, theta=0..2*Pi, coordinates=polar[r,theta], output=integral);from which we see that Maple correctly maps the element of area NiMvSSNkQUc2IiomSSNkeUdGJSIiIkkjZHhHRiVGKA== = NiMqJkkjZHhHNiIiIiJJI2R5R0YlRiY= to NiMqJkkickc2IiIiIkkjZHJHRiVGJg== dNiNJJnRoZXRhRzYi.<Text-field layout="Heading 1" style="Heading 1">The Modified <Font italic="true">int</Font> Command in the <Font italic="true">VectorCalculus</Font> Package</Text-field>The VectorCalculus package modifies the top-level int command to give it the ability to write and evaluate iterated integrals of scalars, and single integrals of vectors. To see just how int performs in this package, executerestart;
interface(warnlevel=0):
with(VectorCalculus):
BasisFormat(false):to access the package. (The BasisFormat command causes all vectors to be written as column vectors.)Then, an iterated integral such asNiMtSSRJbnRHNiI2JC1GJDYkLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YtOy0mRi02IyIiIjYjRiwtJkYtNiMiIiNGNC9GLDsmRixGMiZGLEY3orNiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUYkNiQtRiQ2JC1JImZHRig2JUkieEdGKEkieUdGKEkiekdGKC9GMzstJkYzNiMiIiI2JEYxRjItJkYzNiMiIiNGOi9GMjstJkYyRjg2I0YxLSZGMkY9RkMvRjE7JkYxRjgmRjFGPQ==is implemented with the syntaxint(f(x,y), [x,y]=Region(x1..x2, y1(x)..y2(x)), inert);orint(f(x,y,z), [x,y,z]=Region(x1..x2, y1(x)..y2(x), z1(x,y)..z2(x,y)), inert);Note that it is not possible to use subscripted variables in this instance because, for example, NiMmSSJ4RzYiNiMiIiI= is seen as the variable NiNJInhHNiI=. Note also that the inert form of the integral is generated by using either the parameter "inert" or "output=integral." (The MultiInt command does not accept "inert".) Finally, note that the order of entry for the bounds on the integrals corresponds to operator notation, and is therefore the opposite of the "natural" order described above.In addition to being able to integrate over general regions with a single call to int, it is possible to integrate over special pre-defined regions such as circles, spheres, triangles, rectangles, and ellipses. The following is a useful summary of such functionality. int(f, [x,y] = Circle(<a,b>,r))
int(f, [x,y] = Ellipse(x^2/a^2 + y^2/b^2 -1))
int(f, [x,y] = Rectangle(a..b, c..d)) int(f, [x,y] = Triangle(<a,b>, <c,d>, <p,q>)) int(f, [x,y] = Sector(Circle(<a,b>,r), theta[1], theta[2]) int(f, [x,y] = Sector(Ellipse(x^2/a^2 + y^2/b^2 - 1), theta[1], theta[2]) int(f, [x,y,z] = Sphere(<a,b,c>, r))
int(f, [x,y,z] = Parallelepiped(a..b, c..d, p..q)) int(f, [x,y,z] = Tetrahedron(<a1, b1, c1>, <a2, b2, c2>, <a3, b3, c3>, <a4, b4. c4>))To illustrate integration off := x*y;over the triangle whose vertices are (0, 0), (3, 5), and (2, 4), useint(f, [x,y] = Triangle(<0,0>, <3,5>, <2,4>));To obtain this result without the special properties of the modified int command, begin by finding the equations of the lines between the vertices. These equations areY12 := rhs(Student[Precalculus][Line]([0,0],[3,5])[1]);
Y23 := rhs(Student[Precalculus][Line]([3,5],[2,4])[1]);
Y31 := rhs(Student[Precalculus][Line]([2,4],[0,0])[1]);A graph of the region of integration is contained in Figure 1.plot([[x,Y12,x=0..3],[x,Y23,x=2..3],[x,Y31,x=0..2]], color=[black,red,green], scaling=constrained, title="Figure 1");The integration over the triangle must be written as the sum of the two iterated integralsq := Int(Int(f, y=Y12..Y31), x=0..2) + Int(Int(f, y=Y12..Y23), x=2..3);whose value isvalue(q);As a second example, consider the integral off;over the circle with center at (3, 4) and with radius 2, an iterated integral whose value isint(f, [x,y] = Circle(<3,4>, 2));The inert form of this integral is returned asq := int(f, [x,y] = Circle(<3,4>, 2), inert);where polar coordinates have been used. This is more easily seen if, instead of NiNJInhHNiI= and NiNJInlHNiI=, we cause the integral to be written with the variables NiNJInJHNiI= and NiNJJnRoZXRhRzYi by the brute-force substitutionQ := subs(x=r, y=theta, q);Of course, the integral still evaluates tovalue(Q);<Text-field layout="Heading 1" style="Heading 1">The <Font italic="true">SurfaceInt</Font> Command in the <Font italic="true">VectorCalculus</Font> Package</Text-field>The surface integral of a scalar function is implemented with the SurfaceInt command in the VectorCalculus package. The surface integral of NiMtSSJnRzYiNiVJInhHRiVJInlHRiVJInpHRiU= over that portion of the surface NiMvSSJ6RzYiLUYkNiRJInhHRiVJInlHRiU= inside the cylinder with footprint NiNJIlJHNiI=, the region bounded by the curves NiMvSSJ5RzYiLSZGJDYjIiIiNiNJInhHRiU=, NiMvSSJ5RzYiLSZGJDYjIiIjNiNJInhHRiU=, NiMvSSJ4RzYiSSJhR0Yl, and NiMvSSJ4RzYiSSJiR0Yl, is given bySurfaceInt(g(x,y,z), [x,y,z]=Surface(<x,y,z(x,y)>, x=a..b, y=y1(x)..y2(x)),inert);Recognizing the surface-area element asdNiMvSSZzaWdtYUc2IiooLUklc3FydEc2JEkqcHJvdGVjdGVkR0YqSShfc3lzbGliR0YlNiMsKCIiIkYuKiQmSSJ6R0YlNiNJInhHRiUiIiNGLiokJkYxNiNJInlHRiVGNEYuRi5JI2R5R0YlRi5JI2R4R0YlRi4=we see that this integral is just the "volume" under the surface NiMqJkkiZ0c2IiIiIi1JJXNxcnRHNiRJKnByb3RlY3RlZEdGKkkoX3N5c2xpYkdGJTYjLChGJkYmKiQmSSJ6R0YlNiNJInhHRiUiIiNGJiokJkYwNiNJInlHRiVGM0YmRiY= but inside the cylinder whose footprint in the xy-plane is the region NiNJIlJHNiI=. For example, it we take NiMvLUkiZ0c2IjYkSSJ4R0YmSSJ5R0YmIiIi and the surface NiMvSSJ6RzYiIiIi with NiNJIlJHNiI= bounded by the curves NiMvJkkieUc2IjYjIiIiKiRJInhHRiYiIiM= and NiMvJkkieUc2IjYjIiIjSSJ4R0Ym, the surface integral is just the area of NiNJIlJHNiI=, given by the SurfaceInt command asSurfaceInt(1,[x,y,z]=Surface(<x,y,1>,x=0..1,y=x^2..x));However, the area of NiNJIlJHNiI= is also given byInt(x-x^2,x=0..1) = int(x-x^2, x=0..1);The SurfaceInt command also computes the surface integral of NiMtSSJnRzYiNiVJInhHRiVJInlHRiVJInpHRiU= when the surface NiMvSSJ6RzYiLUYkNiRJInhHRiVJInlHRiU= lies inside the cylinder whose cross-section NiNJIlJHNiI= is a valid region recognized by the modified int command of the VectorCalculus package. For example, if to compute the surface integral off := x*y*z;on that portion of the surfaceZ := 16 - x^2 - y^2;lying inside the cylinder whose cross-section is a circle with center (2, 3) and radius 1, useq := evalf(SurfaceInt(f, [x,y,z]=Surface(<x,y,Z>, [x,y]=Circle(<2,3>,1)), inert));The surface integral of NiMtSSJnRzYiNiVJInhHRiVJInlHRiVJInpHRiU= over a surface described parametrically byNiMvSSJ4RzYiLUYkNiRJInVHRiVJInZHRiU=NiMvSSJ5RzYiLUYkNiRJInVHRiVJInZHRiU=NiMvSSJ6RzYiLUYkNiRJInVHRiVJInZHRiU=NiMxSSJhRzYiSSJ1R0YlNiMxSSFHNiJJImJHRiU=NiMxSSJjRzYiSSJ2R0YlNiMxSSFHNiJJImRHRiU=is given bySurfaceInt(g(x,y,z), [x,y,z]=Surface(<x(u,v),y(u,v),z(u,v)>, u=a..b, v=c..d),inert);The parametric form for the surface-area element when the surface is given parametrically isdNiMvSSZzaWdtYUc2Ii1JJXNxcnRHNiRJKnByb3RlY3RlZEdGKUkoX3N5c2xpYkdGJTYjLCgqJCZJIkpHRiU2IyIiIiIiI0YxKiQmRi82I0YyRjJGMSokJkYvNiMiIiRGMkYxNiMqJkkjZHVHNiIiIiJJI2R2R0YlRiY=where NiM2JSZJIkpHNiI2IyIiIiZGJTYjIiIjJkYlNiMiIiQ= are, respectively, the JacobiansNiMmSSJKRzYiNiMiIiI= = det NiMtSSdNYXRyaXhHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjNyQ3JCZJInlHRig2I0kidUdGKCZGLTYjSSJ2R0YoNyQmSSJ6R0YoRi4mRjVGMQ== = NiMsJiomJkkieUc2IjYjSSJ1R0YnIiIiJkkiekdGJzYjSSJ2R0YnRipGKiomJkYmRi1GKiZGLEYoRiohIiI=NiMmSSJKRzYiNiMiIiM= = det NiMtSSdNYXRyaXhHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjNyQ3JCZJInpHRig2I0kidUdGKCZGLTYjSSJ2R0YoNyQmSSJ4R0YoRi4mRjVGMQ== = NiMsJiomJkkiekc2IjYjSSJ1R0YnIiIiJkkieEdGJzYjSSJ2R0YnRipGKiomJkYmRi1GKiZGLEYoRiohIiI=NiMmSSJKRzYiNiMiIiQ= = det NiMtSSdNYXRyaXhHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjNyQ3JCZJInhHRig2I0kidUdGKCZGLTYjSSJ2R0YoNyQmSSJ5R0YoRi4mRjVGMQ== = NiMsJiomJkkieEc2IjYjSSJ1R0YnIiIiJkkieUdGJzYjSSJ2R0YnRipGKiomJkYmRi1GKiZGLEYoRiohIiI=which can be obtained in Maple via the syntaxJacobian([y(u,v),z(u,v)],[u,v], determinant);
Jacobian([z(u,v),x(u,v)],[u,v], determinant);
Jacobian([x(u,v),y(u,v)],[u,v], determinant);If a sphere of radius NiNJImFHNiI= > 0 is described in spherical coordinates by NiMxIiIhSSRwaGlHNiI=NiMxSSFHNiJJI1BpR0kqcHJvdGVjdGVkR0Yn, NiMxIiIhSSZ0aGV0YUc2Ig==NiMxSSFHNiIqJiIiIyIiIkkjUGlHSSpwcm90ZWN0ZWRHRipGKA==, then its surface area is given bySurfaceInt(1, [x,y,z] = Surface(<a,phi,theta>, phi=0..Pi, theta=0..2*Pi, coords=spherical)) assuming a>0;Alternatively, the surface area of a sphere of radius NiNJImFHNiI= could also be found withSurfaceInt(1, [x,y,z] = Sphere(<0,0,0>, a ));In the first calculation, the parametrization was in terms of spherical coordinates, while in the second, Maple understood one of two pre-defined surfaces. The second such surface is the box.To obtain the surface integral of the scalar NiMtSSJnRzYiNiVJInhHRiVJInlHRiVJInpHRiU= over the surface of a rectangular box, the syntaxSurfaceInt(g(x,y,z), [x,y,z] = Box(a..b, c..d, r..s), inert);can be used. Each of the three integrals returned by Maple combines the contributions from opposite faces of the box.<Text-field layout="Heading 1" style="Heading 1">The <Font italic="true">Flux</Font> Command in the <Font italic="true">VectorCalculus</Font> Package</Text-field>The Flux command in the VectorCalculus package will compute the flux (surface integral of the normal component of the field) of a vector field through a surface. If the surface is closed, the user has the choice of implementing the flux integral with either the inward or outward normal. Thus, the flux of the vector fieldF := VectorField(<x,y,z>/sqrt(x^2+y^2+z^2), cartesian[x,y,z]);through the surface of the unit sphere centered at the origin can be computed as either ofFlux(F, Sphere(<0,0,0>,1));
Flux(F, Sphere(<0,0,0>,1, inward));In the first case, the outward normal is used, but in the second, the inward normal is used.No such control over the normal is provided for an open surface. Maple arbitrarily picks a normal direction and provides neither for changing its orientation, nor for discovering which orientation was used. Thus, to integrate the vector field F over the surface of the upper hemisphere, useFlux(F, Surface(<a,phi,theta>, phi=0..Pi/2, theta=0..2*Pi, coords=spherical)) assuming a>0;From the value returned, it is obvious that the upward normal (outward on the closed sphere) was used. But in general, it would be difficult to tell which way the net flux was directed for an arbitrary field and (open) surface.A calculation of the flux of F through the upper hemisphere can be obtained from first principles in which the direction of the normal would be known. From the value of the flux obtained this way, it would be possible to infer the direction of the normal field used by Maple's Flux command.To this end, describe the upper hemisphere explicitly withZ := sqrt(a^2 - x^2 - y^2);so that a normal field on the surface is given byN1 := <-diff(Z,x), -diff(Z,y), 1>;To obtain a unit normal field, the Normalize command from the LinearAlgebra package is needed. Hence, executewith(LinearAlgebra):and thenN2 := Normalize(N1,2);Of course, this calculation is valid in Cartesian coordinates only.Further simplification of the unit normal field is obtained withN := simplify(N2, symbolic);from which it's clear that the normal is upward (outward on the closed sphere).To obtain F . N, useFN := simplify(DotProduct(F,N, conjugate=false));where the DotProduct command is the one in the LinearAlgebra package.The flux of F through the upper hemisphere is the surface integral of F . N, obtained in Maple withq := SurfaceInt(FN, [x,y,z]=Surface(<x,y,Z>, x=-a..a, y=-sqrt(a^2-x^2)..sqrt(a^2-x^2)));To simplify the integrand of this integral, extract is viaf := op([1,1],q);and simplify it viaf1 := simplify(f) assuming a>0The surface integral is merely the integral of this quantity over a disk with center at the origin, with radius NiNJImFHNiI=, and lying in the xy-plane. This integral can be obtained in Maple withint(f1,[x,y]=Circle(<0,0>,a)) assuming a>0;which agrees with the result of the Flux command. Hence, the Flux command used the upward (outward on the closed sphere) normal, as we did in our calculation from first principles.Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft.
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