Multiple_Integrals.mth: Solving problems of Multiple Integrals with ...

Dresden International Symposium on Technology and its
Integration into Mathematics Education 2006
DES-TIME 2006
DRESDEN 20th-23th July 2006
Multiple_Integrals.mth: Solving
problems of Multiple Integrals
with Derive 6
G. Aguilera C. Cielos J. L. Galán
M. A. Galán A. Gálvez A. J. Jiménez
Y. Padilla P. Rodríguez
Dpt. Applied Mathematic
University of Málaga (Spain)

Introduction.
Our experience:
– Software to use.
Index
– Innovative aspect of the experience.
Multiple Integrals with Derive.
Final conclusions.

Introduction
In most cases, the use of CAS is reduced
to using computers as powerful highperformance
calculators.
It is therefore necessary to change the
way people think about information
technologies in order to optimise the
opportunities they offer and try to
encourage mathematical creativity among
students.

Introduction
Math teachers that use CAS have to change the
traditional uses given to these tools.
It is a mistake to use CAS in teaching as simple
problem-solving machines.
They should be used in ways that maximize the
opportunities that these technologies offer:
positively affecting student learning, significantly
increasing opportunities for experimentation and
allowing students to construct their own
mathematical knowledge.

Introduction
Math teachers must first set out a series
of appropriate activities.
The use of CAS in Mathematics has not
reached optimum conditions. Mostly are
blackbox (showing the result in one step
without teaching students how to get
there) and should be whitebox (showing
intermediate steps).

Combining programming
and CAS
When students program, they must read,
construct and refine strategies, modify
previously written programs and lastly, use the
programs to solve problems. This makes them
the protagonists of their own learning.
The most appropriate approach involves using
programming and CAS together to allow
students to create the specific necessary
functions that will allow them to solve the
problems involved in the subject matter under
study.

Our experience
Our experience over the past ten years
has consisted mainly of continuously
carrying out practical experiments in the
computer laboratory in Mathematics
courses for the degrees of Technical
Industrial Engineering and Technical
Telecommunications Engineering at the
University of Málaga.

Software to use
Among the great amount of mathematical
software, we have chosen the program
Derive for several reasons:
1. Derive is easier to use than other mathematical
programs as it operates with a very simple syntax.
2. The student is capable of starting to solve
problems by using the program in a short period of
time, since basic functions and operations are
available in several menus
3. It needs few requirements, with regard either to
memory and physical space, when it comes to
installing it

Innovative aspect of our
experience
The main innovative aspect of these
experiences is that the students have an
active role in the sense they should
elaborate themselves utility files to solve
the typical problems for the different
subjects. This fact implies that the
students need to deal with programming
in Derive, to understand the subject and
to know how to solve typical problems.

Innovative aspect of our
experience
The fact that the students themselves
are making the programs and including
all of their arguments has a positive
influence on their ability to subsequently
apply them to solve specific exercises.
So, with this kind of activities CAS
encourages active learning, greater
comprehension of subject matter and
mathematical creativity among students.

Double integrals:
– In Cartesian coordinates.
– In polar coordinates.
Triple integrals:
Contents of
Multiple_Integrals.mth
– In Cartesian coordinates
– In cylindrical coordinates.
– In spherical coordinates.
Applications of Multiple Integrals

Final Conclusions
Our accumulated experience reveals that
CAS are computer tools which are easy to
use and useful in Mathematics courses for
Engineering.
The traditional uses given to CAS in the
teaching of Mathematics for Engineering must
be changed to maximize the opportunities
offered by CAS technologies. Optimal use
should be aimed at improving student
motivation, autonomy and achieving
participatory and student-centred learning.

Final Conclusions
One powerful idea involves combining CAS
resources with the flexibility of a programming
language.
There exists reasonable evidence to show
that making programs with Derive facilitates
learning and improves student motivation.
Although it would be desirable to do so, it is
not necessary to substantially modify the
traditional program of studies of Math courses
for Engineering to introduce the innovation of
having students make their own programs
with Derive.

Dresden International Symposium on Technology and its
Integration into Mathematics Education 2006
DES-TIME 2006
DRESDEN 20th-23th July 2006
Multiple_Integrals.mth: Solving
problems of Multiple Integrals
with Derive 6
G. Aguilera C. Cielos J. L. Galán
M. A. Galán A. Gálvez A. J. Jiménez
Y. Padilla P. Rodríguez
Dpt. Applied Mathematic
University of Málaga (Spain)

________________________________________________________
Multiple_Integrals.dfw:
Multiple integrals and its applications
July, 2006
José Luis Galán García
Pedro Rodríguez Cielos
M. Angeles Galán García
Yolanda Padilla Domínguez
Department of Applied Mathematic
University of Málaga
http://www.satd.uma.es/matap/
________________________________________________________
The following functions have been developed in this utility/demo dfw file to deal with multiple integrals and some of
their applications:
• Doble and triple integrals. Green-Riemann's theorem
o double(f,var1,lim1,lim2,var2,lim3,lim4) to calculate the double integral of f with the
corresponding variables and limits of integration using
cartesian coordinates.
o doublepolar(f,º,º1,º2,²,²1,²2) to calculate the double integral of f with the
corresponding variables and limits of integration using
polar coordinates.
o triple(f,var1,lim1,lim2,var2,lim3,lim4,var3,lim5,lim6) to calculate the triple integral of f with the
corresponding variables and limits of integration using
cartesian coordinates.
o triplecylindrical(f,z,z1,z2,º,º1,º2,²,²1,²2) to calculate the triple integral of f with the corresponding
variables and limits of integration using cylindrical
coordinates.
o triplespherical(f,º,º1,º2,²,²1,²2,¾,¾1,¾2) to calculate the triple integral of f with the corresponding
variables and limits of integration using spherical
coordinates.
• Applications: Green-Riemann's theorem. Surface integrals. Gauss' theorem
o linegreenriemann(P,Q,y,y1,y2,x,x1,x2) to calculate the line integral of the field (P,Q) along the
curve limit of the region bounded by the corresponding
variables and limits of integration in cartesian coordinates,
using Green-Riemann's theorem.

o linegreenriemannpolar(P,Q,º,º1,º2,²,²1,²2) to calculate the line integral of the field (P,Q) along the
curve limit of the region bounded by the corresponding
variables and limits of integration in polar coordinates,
using Green-Riemann's theorem.
o surfacearearxy(S,y,y1,y2,x,x1,x2) to calculate the area of the surface S ≡ z=z(x,y) with the limits of
integration in Rxy.
o surfacearearxz(S,x,x1,x2,z,z1,z2) to calculate the area of the surface S ≡ y=y(x,z) with the limits of
integration in Rxz.
o surfacearearyz(S,z,z1,z2,y,y1,y2) to calculate the area of the surface S ≡ x=x(y,z) with the limits of
integration in Ryz.
o surfacearearxypolar(S,º,º1,º2,²,²1,²2) to calculate the area of the surface S ≡ z=z(x,y) with the limits of
integration in Rxy in polar coordinates.
o surfacearearxzpolar(S,º,º1,º2,²,²1,²2) to calculate the area of the surface S ≡ y=y(x,z) with the limits of
integration in Rxz in polar coordinates.
o surfacearearyzpolar(S,º,º1,º2,²,²1,²2) to calculate the area of the surface S ≡ x=x(y,z) with the limits of
integration in Ryz in polar coordinates.
o surfacerxy(f,S,y,y1,y2,x,x1,x2) to integrate the function f over the surface S ≡ z=z(x,y) with the
limits of integration in Rxy.
o surfacerxz(f,S,x,x1,x2,z,z1,z2) to integrate the function f over the surface S ≡ y=y(x,z) with the
limits of integration in Rxz.
o surfaceryz(f,S,z,z1,z2,y,y1,y2) to integrate the function f over the surface S ≡ x=x(y,z) with the
limits of integration in Ryz.
o surfacerxypolar(f,S,º,º1,º2,²,²1,²2) to integrate the function f over the surface S ≡ z=z(x,y) with the
limits of integration in Rxy in polar coordinates.
o surfacerxzpolar(f,S,º,º1,º2,²,²1,²2) to integrate the function f over the surface S ≡ y=y(x,z) with the
limits of integration in Rxz in polar coordinates.
o surfaceryzpolar(f,S,º,º1,º2,²,²1,²2) to integrate the function f over the surface S ≡ x=x(y,z) with the
limits of integration in Ryz in polar coordinates.
o unitnormalvector(F) to calculate an unit normal vector of the surface S ≡ F(x,y,z)=0.
o fluxrxy(P,Q,R,S,y,y1,y2,x,x1,x2) to calculate the flux of (P,Q,R) over the surface S ≡ z=z(x,y)
with the limits of integration in Rxy.
o fluxrxz(P,Q,R,S,x,x1,x2,z,z1,z2) to calculate the flux of (P,Q,R) over the surface S ≡ y=y(x,z)
with the limits of integration in Rxz.
o fluxryz(P,Q,R,S,z,z1,z2,y,y1,y2) to calculate the flux of (P,Q,R) over the surface S ≡ x=x(y,z)
with the limits of integration in Ryz.
o fluxrxypolar(P,Q,R,S,º,º1,º2,²,²1,²2) to calculate the flux of (P,Q,R) over the surface S ≡ z=z(x,y)
with the limits of integration in Rxy in polar coordinates.
o fluxrxzpolar(P,Q,R,S,º,º1,º2,²,²1,²2) to calculate the flux of (P,Q,R) over the surface S ≡ y=y(x,z)
with the limits of integration in Rxz in polar coordinates.
o fluxryzpolar(P,Q,R,S,º,º1,º2,²,²1,²2) to calculate the flux of (P,Q,R) over the surface S ≡ x=x(y,z)
with the limits of integration in Ryz in polar coordinates.
o fluxgauss(P,Q,R,z,z1,z2,y,y1,y2,x,x1,x2) to calculate the flux of (P,Q,R) over the surface S limit of the
solid which limits of integration are given in cartesian
coordinates.
o fluxgausscylindrical(P,Q,R,z,z1,z2,º,º1,º2,²,²1,²2) to calculate the flux of (P,Q,R) over the surface S
limit of the solid which limits of integration are
given in cylindrical coordinates.

o fluxgaussspherical(P,Q,R,º,º1,º2,²,²1,²2,¾,¾1,¾2) to calculate the flux of (P,Q,R) over the surface S
limit of the solid which limits of integration are
given in spherical coordinates.
MULTIPLE INTEGRALS. GREEN-RIEMANN'S THEOREM
_____________________________________________________
• Double integration in cartesian coordinates
o Syntax: DOUBLE(function,var1,lim1,lim2,var2,lim3,lim4)
o Example: double(xy,y,0,x+1,x,0,2) to integrate the function f(x,y) = xy
in the region bounded by
x = 2 , y = x+1 , y = 0 and x = 0
#1: DOUBLE(x·y, y, 0, x + 1, x, 0, 2)
17
#2: ⎯⎯
3
• Double integration in polar coordinates
o Syntax: DOUBLEPOLAR(function,r,r1,r2,theta,theta1,theta2)
o Example: doublepolar(x^2y^2/(x^2+y^2),r,0,1,theta,0,pi)
to integrate the function f(x,y) = x 2 y 2 /(x 2 + y 2 ) in the region
bounded by x 2 +y 2 = 1 with y ’ 0
⎛ 2 2 ⎞
⎜ x ·y ⎟
#3: DOUBLEPOLAR⎜⎯⎯⎯⎯⎯⎯⎯, r, 0, 1, θ, 0, π⎟
⎜ 2 2 ⎟
⎝ x + y ⎠
π
#4: ⎯⎯
32

• Triple integration in cartesian coordinates
o Syntax: TRIPLE(function,var1,lim1,lim2,var2,lim3,lim4,var3,lim5,lim6)
o Example: triple(x+yz,z,0,5-x-y,y,0,5-x,x,0,5) to integrate the function
f(x,y,z) = x+yz in the solid
bounded by x+y+z = 5 ,
x = 0 , y = 0 and z = 0
#5: TRIPLE(x + y·z, z, 0, 5 - x - y, y, 0, 5 - x, x, 0, 5)
625
#6: ⎯⎯⎯
12
• Triple integration in cylindrical coordinates
o Syntax: TRIPLECYLINDRICAL(function,z,z1,z2,r,r1,r2,theta,theta1,theta2)
o Example: triplecylindrical(sqrt(x^2+y^2),z,r,1,r,0,1,theta,0,2pi)
to integrate the function f(x,y,z) = ‹(x 2 +y 2 ) in the solid
bounded by z = 0 , z = 1 and z 2 = x 2 +y 2
2 2
#7: TRIPLECYLINDRICAL(√(x + y ), z, r, 1, r, 0, 1, θ, 0, 2·π)
π
#8: ⎯
6
• Triple integration in spherical coordinates
o Syntax: TRIPLESPHERICAL(function,r,r1,r2,theta,theta1,theta2,phi,phi1,phi2)
o Example: triplespherical(x+y+z,r,0,2,theta,0,2pi,phi,0,pi/2)
to integrate the function f(x,y,z) = x+y+z in the solid
bounded by x 2 + y 2 + z 2 = 4 with z ’ 0
⎛ π ⎞
#9: TRIPLESPHERICAL⎜x + y + z, r, 0, 2, θ, 0, 2·π, α, 0, ⎯⎟
⎝ 2 ⎠

#10: 4·π
APPLICATIONS OF MULTIPLE INTEGRLAS
_________________________________________
• Green-Riemann's theorem in cartesian coordinates
o Syntax: LINEGREENRIEMANN(comp1,comp2,y,y1,y2,x,x1,x2)
o Example: linegreenriemann(y^2,x,y,0,3,x,0,3) to calculate the line integrate of
(y 2 ,x) along the square of vertices
(0,0) , (3,0) , (3,3) , (0,3) using
Green-Riemann's theorem
2
#11: LINEGREENRIEMANN(y , x, y, 0, 3, x, 0, 3)
#12: -18
• Green-Riemann's theorem in polar coordinates
o Syntax: LINEGREENRIEMANNPOLAR(comp1,comp2,r,r1,r2,theta,theta1,theta2)
o Example: linegreenriemannpolar(y^2,x,r,0,5,theta,0,2pi)
to calculate the line integrate of (y 2 ,x) along the circumference
x 2 +y 2 =25 using Green-Riemann's theorem
2
#13: LINEGREENRIEMANNPOLAR(y , x, r, 0, 5, θ, 0, 2·π)
#14: 25·π
• Surface area in polar coordinates
o Syntax: SURFACEAREARXYPOLAR(z surface,r,r1,r2,theta,theta1,theta2)

o Example: surfacearearxypolar(sqrt(x^2+y^2),r,0,1,theta,0,2pi)
to calculate the area of the part of the surface z 2 = x 2 + y 2
inside of z = 2 - x 2 - y 2 with z ’ 0
2 2
#15: SURFACEAREARXYPOLAR(√(x + y ), r, 0, 1, θ, 0, 2·π)
#16: √2·π
• Unit normal vector to an implicit surface
o Syntax: UNITNORMALVECTOR(implicit surface)
o Example: unitnormalvector(x^2+y^2+z^2-4) to find an unit normal
vector to x 2 + y 2 + z 2 = 4
2 2 2
#17: UNITNORMALVECTOR(x + y + z - 4)
⎡ x y z ⎤
⎢⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯, ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯, ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎥
#18: ⎢ 2 2 2 2 2 2 2 2 2 ⎥
⎣ √(x + y + z ) √(x + y + z ) √(x + y + z ) ⎦
• Flux of a vector field. Double integral in polar coordinates
o Syntax: FLUXRXYPOLAR(comp1,comp2,comp3,zsurface,r,r1,r2,theta,theta1,theta2)
o Example: fluxrxypolar(x,2y,x+z,x^2+y^2,r,0,4,theta,0,2pi)
to calculate the flux of (x,2y,x+z) over the part of the surface
of the paraboloid z = x 2 + y 2 for which 0 ≤ z ≤ 16
2 2
#19: FLUXRXYPOLAR(x, 2·y, x + z, x + y , r, 0, 4, θ, 0, 2·π)
#20: - 256·π
• Gauss' theorem. Triple integral in cartesian coordinates
o Syntax: FLUXGAUSS(c1,c2,c3,z,z1,z2,y,y1,y2,x,x1,x2)

o Example: fluxgauss(x,y,z,z,0,1,y,0,1,x,0,1) to calculate the flux of (x,y,z)
over the closed surface
bounded by the unit cube
#21: FLUXGAUSS(x, y, z, z, 0, 1, y, 0, 1, x, 0, 1)
#22: 3
• Gauss' theorem. Triple integral in cylindrical coordinates
o Syntax: FLUXGAUSSCYLINDRICAL(c1,c2,c3,z,z1,z2,r,r1,r2,theta,theta1,theta2)
o Example: fluxgausscylindrical(x,2y,3z,z,r,2,r,0,2,theta,0,2pi)
to calculate the flux of (x,2y,3z) over the closed surface
bounded by the cone z 2 = x 2 + y 2 and the planes z = 0, z = 2
#23: FLUXGAUSSCYLINDRICAL(x, 2·y, 3·z, z, r, 2, r, 0, 2, θ, 0, 2·π)
#24: 16·π
• Gauss' theorem. Triple integral in spherical coordinates
o Syntax: FLUXGAUSSSPHERICAL(c1,c2,c3,r,r1,r2,theta,theta1,theta2,phi,phi1,phi2)
o Example: fluxgaussspherical(x^2z+x,y,z,r,0,1,theta,0,2pi,phi,0,pi/2)
to calculate the flux of (x 2 z+x,y,z) over the closed surface bounded
by the semi-sphere x 2 + y 2 + z 2 = 1 and z = 0 with z ’ 0
⎛ 2 π ⎞
#25: FLUXGAUSSSPHERICAL⎜x ·z + x, y, z, r, 0, 1, θ, 0, 2·π, φ, 0, ⎯⎟
⎝ 2 ⎠
#26: 2·π

Cartesian coordinates
#1: [u1 ≔, u2 ≔, v1 ≔, v2 ≔, w1 ≔, w2 ≔]
#2: f
u2
#3: ∫ f du
u1
v2
⌠ u2
#4: ⎮ ∫ f du dv
⌡ u1
v1
Double Integrals
v2
⌠ u2
#5: double(f, u, u1, u2, v, v1, v2) ≔ ⎮ ∫ f du dv
⌡ u1
v1
Polar coordinates
#6: SUBST(f, [x, y], [ρ·COS(θ), ρ·SIN(θ)])
#7: ρ·SUBST(f, [x, y], [ρ·COS(θ), ρ·SIN(θ)])
#8: doublepolar(f, u, u1, u2, v, v1, v2) ≔ double(ρ·SUBST(f, [x, y],
[ρ·COS(θ), ρ·SIN(θ)]), u, u1, u2, v, v1, v2)

Cartesian coordinates
#9: f
u2
#10: ∫ f du
u1
v2
⌠ u2
#11: ⎮ ∫ f du dv
⌡ u1
v1
w2
⌠ v2
⎮ ⌠ u2
#12: ⎮ ⎮ ∫ f du dv dw
⎮ ⌡ u1
⌡ v1
w1
Triple Integrals
w2
⌠ v2
⎮ ⌠ u2
#13: triple(f, u, u1, u2, v, v1, v2, w, w1, w2) ≔ ⎮ ⎮ ∫ f du dv dw
⎮ ⌡ u1
⌡ v1
w1
Cylindrical coordinates
#14: SUBST(f, [x, y, z], [ρ·COS(θ), ρ·SIN(θ), z])
#15: ρ·SUBST(f, [x, y, z], [ρ·COS(θ), ρ·SIN(θ), z])
#16: triplecylindrical(f, u, u1, u2, v, v1, v2, w, w1, w2) ≔
triple(ρ·SUBST(f, [x, y, z], [ρ·COS(θ), ρ·SIN(θ), z]), u, u1,
u2, v, v1, v2, w, w1, w2)

Spherical coordinates
#17: SUBST(f, [x, y, z], [ρ·COS(φ)·COS(θ), ρ·COS(φ)·SIN(θ), ρ·SIN(φ)])
2
#18: ρ ·COS(φ)·SUBST(f, [x, y, z], [ρ·COS(φ)·COS(θ), ρ·COS(φ)·SIN(θ),
ρ·SIN(φ)])
#19: triplespherical(f, u, u1, u2, v, v1, v2, w, w1, w2) ≔
2
triple(ρ ·COS(φ)·SUBST(f, [x, y, z], [ρ·COS(φ)·COS(θ),
ρ·COS(φ)·SIN(θ), ρ·SIN(φ)]), u, u1, u2, v, v1, v2, w, w1, w2)
Exercise 1
Exercises
Compute the area of the circle x^2+y^2 ≤ 16 using Cartesian
and polar coordinates.
2 2
#20: x + y ≤ 16

2 2
#21: double(1, y, - √(16 - x ), √(16 - x ), x, -4, 4)
#22: 16·π
2 2
#23: SUBST(x + y ≤ 16, [x, y], [ρ·COS(θ), ρ·SIN(θ)])
#24: -4 ≤ ρ ≤ 4
#25: doublepolar(1, ρ, 0, 4, θ, 0, 2·π)
#26: 16·π
Exercise 2
Calculate the volume of the solid bounded below by the cone
z = √(x^2+y^2) and above by the sphere x^2+y^2+z^2 = 2
using Cartesian, cylindrical and spherical coordinates.
2 2
#27: z = √(x + y )
2 2
#28: z = √(2 - x - y )
2 2 2 2 2
#29: triple(1, z, √(x + y ), √(2 - x - y ), y, - √(1 - x ), √(1 -
2
x ), x, -1, 1)

#30: The computer cannot compute this calculation in Cartesian
coordinates.
2 2
#31: z = √((ρ·COS(θ)) + (ρ·SIN(θ)) )
#32: z = ⎮ρ⎮
2 2
#33: z = √(2 - (ρ·COS(θ)) - (ρ·SIN(θ)) )
2
#34: z = √(2 - ρ )
2
#35: triplecylindrical(1, z, ρ, √(2 - ρ ), ρ, 0, 1, θ, 0, 2·π)
⎛ 4·√2 4 ⎞
#36: π·⎜⎯⎯⎯⎯ - ⎯⎟
⎝ 3 3 ⎠
2 2 2
#37: z = x + y
2 2 2
#38: (ρ·SIN(φ)) = (ρ·COS(φ)·COS(θ)) + (ρ·COS(φ)·SIN(θ))
2 2 2 2
#39: ρ ·SIN(φ) = ρ ·COS(φ)
π
#40: φ = ⎯
4
2 2 2
#41: x + y + z = 2
2 2 2
#42: (ρ·COS(φ)·COS(θ)) + (ρ·COS(φ)·SIN(θ)) + (ρ·SIN(φ)) = 2
2
#43: ρ = 2
⎛ π π ⎞
#44: triplespherical⎜1, ρ, 0, √2, θ, 0, 2·π, φ, ⎯, ⎯⎟
⎝ 4 2 ⎠
⎛ 4·√2 4 ⎞
#45: π·⎜⎯⎯⎯⎯ - ⎯⎟
⎝ 3 3 ⎠