Line_Integrals.mth: Solving problems of Line Integrals with Derive 6

Dresden International Symposium on Technology and its
Integration into Mathematics Education 2006
DES-TIME 2006
DRESDEN 20th-23th July 2006
Line_Integrals.mth: Solving
problems of Line 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.
Line_Integrals.mth.
Final conclusions.
Index

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.

Parametrization of curves.
Exact differential forms.
Potential function.
Line Integrals.
Contents of
Line_Integrals.mth
Line Integrals with respect to arc length.
Applications of Line Integrals.

We prefer to use large names for macros
instead of using shorthands because we
think it is easier for pupils to remember the
full name and, on the other hand, the large
name of the macro allows to guess what it
solves

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
Line_Integrals.mth: Solving
problems of Line 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)

_________________________________________________________
Line Integrals.dfw
July, 2006
Jose Luis Galan Garcia (jl_galan@uma.es)
Pedro Rodriguez Cielos (prodriguez@uma.es)
M. Angeles Galan Garcia (magalan@ctima.uma.es)
Yolanda Padilla Dominguez (ypadilla@ctima.uma.es)
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 line integrals and some of their
applications:
• Parametrization of curves
o Segment(p1,p2) parametrization of the segment which joins p1 with p2.
o Circumference(x0,y0,r) parametrization of the circumference (x-x0)^2 + (y-y0)^2 = r^2.
o Ellipse(x0,y0,a,b) parametrization of the ellipse (x-x0)^2/a^2 + (y-y0)^2/b^2 = 1.
o Lemniscata(a) parametrization of the lemniscata (x^2+y^2)^2 = a^2 (x^2-y^2).
o Astroid(a) parametrization of the astroid x^(2/3) + y^(2/3) = a^(2/3).
o EllipticalAstroid(a,b) parametrization of the elliptical astroid (x/a)^(2/n) + (y/b)^(2/n) = 1.
o Cardioid(a) parametrization of the cardioid (x^2+y^2-2ax)^2 = 4a^2(x^2+y^2).
o Catenary(a) parametrization of the catenary y = a cosh(x/a).
o Cicloid(r,h) parametrization of the cicloid [rt - h sin(t) , r - h cos(t)].
o Cisoid(a) parametrization of the cisoid x^3+xy^2 = 2ay^2.
o DescartesFolium(a) parametrization of Descartes' folium x^3+y^3 = 3axy.
o EightFigure(a) parametrization of the eight figure x^4 = a^2(x^2-y^2).
o PascalSnail(a,b) parametrization of Pascal's snail (x^2+y^2-2ax)^2 = b^2(x^2+y^2).
o Rosacea(a,k) parametrization of Rosacea [a sin(kt)cos(t), a sin(kt)sin(t)].
o Folium(a,b) parametrization of the folium (x^2+y^2) (x^2+y^2+xb) = 4axy^2.
o Hipocicloid(a,b) parametrization of the hipocicloid
[(a-b)cos(t)+bcos((a-b)t/b),(a-b)sin(t)-bsin((a-b)t/b)].
o Trisectriz(a) parametrization of the trisectriz y^2(a+x) = x^2(3a-x).
o Tractriz(a) parametrization of the tractriz [a sin(t), a (cos(t) + ln(tan(t/2)))].
o ArchimedesSpiral(a) parametrization of Archimedes' spiral [at cos(t), at sin(t)].
o Astroid3D(a,b,n) parametrization of the astroid [a cos^n(t), b sin^n(t), cos(2t)].
o SphericalSpiral(a,n,m) parametrization of the spherical spiral
[a cos(nt)cos(mt), a cos(nt)sin(mt), a sin(nt)].
o VivianiCurve(a) parametrization of Viviani's curve [a(1+cos(t)), a sin(t), 2a sin(t/2)].
o HelicoidalCurve(curve,c) parametrization of the helicoidal curve [x(t), y(t), ct] where curve=[x(t),y(t)].

• Exact differential
o ExactDifferential2(p,q) to check if Pdx + Qdy is an exact differential.
o ExactDifferential3(p,q,r) to check if Pdx + Qdy + Rdz is an exact differential.
o ExactDifferential(F) to check if Fdα is an exact differential.
• Potential function
o Potential2(p,q) to calculate, if there exists, the potential function of P dx + Q dy.
o Potential3(P,Q,R) to calculate, if there exists, the potential function of P dx + Q dy + R dz.
o Potential(F) to calculate, if there exists, the potential function of Fdα.
• Line Integrals
o LineIntegral2(p,q,c1,c2,a,b) to integrate the field (P,Q) along the curve (c1,c2) which parameter t
belongs to [a,b].
o Lineintegral3(p,q,r,c1,c2,c3,a,b) to integrate the field (P,Q,R) along the curve (c1,c2,c3) which parameter t
belongs to [a,b].
o Lineintegral(F,C,a,b) to integrate the field F along the curve C which parameter t belongs to
[a,b].
• Line Integrals with respect to arc length
o ArcLineIntegral2(f,c1,c2,a,b) to integrate the scalar field f along the curve (c1,c2) which parameter t
belongs to [a,b].
o ArcLineintegral3(f,c1,c2,c3,a,b) to integrate the scalar field f along the curve (c1,c2,c3) which parameter t
belongs to [a,b].
o ArcLineintegral(f,C,a,b) to integrate the scalar field f along the curve C which parameter t
belongs to [a,b].
• Applications of Line Integrals
o AreaInsideCurve(C,a,b) to compute the area inside the curve C which parameter t belongs to [a,b].
o CurveLength(C,a,b) to compute the length of the curve C which parameter t belongs to [a,b].
o WireMass(ρ,C,a,b) to compute the mass of a wire of equation C which parameter t belongs to [a,b],
being ρ its density function.
o WireMx(ρ,C,a,b) to compute the static moment with respect to X-axe of a wire of equation C
which parameter t belongs to [a,b], being ρ its density function.
o WireMy(ρ,C,a,b) to compute the static moment with respect to Y-axe of a wire of equation C
which parameter t belongs to [a,b], being ρ its density function.
o WireMxy(ρ,C,a,b) to compute the static moment with respect to XY-plane of a wire of equation C
which parameter t belongs to [a,b], being ρ its density function.
o WireMxz(ρ,C,a,b) to compute the static moment with respect to XZ-plane of a wire of equation C
which parameter t belongs to [a,b], being ρ its density function.
o WireMyz(ρ,C,a,b) to compute the static moment with respect to YZ-plane of a wire of equation C
which parameter t belongs to [a,b], being ρ its density function.
o WireMassCenter(ρ,C,a,b) to compute the mass of a wire of equation C which parameter t belongs to [a,b],
being ρ its density function.
o WireInertiaMoment(ρ,C,a,b)to compute the inertial moment with respect to an axe of distance δ to the wire
of equation C which parameter t belongs to [a,b] it, being ρ its density function.
o MediumValue(f,C,a,b) to compute the medium value of the scalar field f along the curve of equation C
which parameter t belongs to [a,b].

• Parametrization of a segment
Parametrization of curves
o Syntax: Segment(p1,p2)
o Example: Segment([a,b],[c,d])
#1: Segment([a, b], [c, d])
to parametize the segment which join [a,b] with [c,d]
#2: [a·(1 - t) + c·t, b·(1 - t) + d·t]
• Parametrization of a circumference
o Syntax: Circumference(x0,y0,r)
o Example: Circumference(1,-2,4)
#3: Circumference(1, -2, 4)
to parametize the circumference (x-1)^2 + (y+2)^2 = 4^2
#4: [4·COS(t) + 1, 4·SIN(t) - 2]
#5: Circumference(x0, y0, r)
#6: [r·COS(t) + x0, r·SIN(t) + y0]

• Parametrization of an ellipse
o Syntax: Ellipse(x0,y0,a,b)
o Example: Ellipse(1,-2,2,3)
#7: Ellipse(1, -2, 2, 3)
to parametize the ellipse (x-1)^2/2^2 + (y+2)^2/3^2 = 1
#8: [2·COS(t) + 1, 3·SIN(t) - 2]
#9: Ellipse(x0, y0, a, b)
#10: [a·COS(t) + x0, b·SIN(t) + y0]
• Parametrization of a lemniscata
o Syntax: Lemniscata(a)
o Example: Lemniscata(3)
to parametize the lemniscata (x^2+y^2)^2 = 3^2 (x^2-y^2)

#11: Lemniscata(3)
#12: [3·COS(t)·√(COS(2·t)), 3·SIN(t)·√(COS(2·t))]
#13: Lemniscata(a)
#14: [a·COS(t)·√(COS(2·t)), a·SIN(t)·√(COS(2·t))]
• Parametrization of an astroid
#15: Astroid(3)
o Syntax: Astroid(a)
o Example: Astroid(3)
to parametize the astroid x^(2/3) + y^(2/3) = 3^(2/3)
⎡ 3 3⎤
#16: ⎣3·COS(t) , 3·SIN(t) ⎦
#17: Astroid(a)
⎡ 3 3⎤
#18: ⎣a·COS(t) , a·SIN(t) ⎦

• Parametrization of an elliptical astroid
o Syntax: EllipticalAstroid(a,b,n)
o Example: EllipticalAstroid(2,1,3)
#19: EllipticalAstroid(2, 1, 3)
to parametize the elliptical astroid x^(2/3)/2^2 + y^(2/3)/1^1 = 1^(2/3)
⎡ 3 3⎤
#20: ⎣2·COS(t) , SIN(t) ⎦
#21: EllipticalAstroid(a, b, n)
⎡ n n⎤
#22: ⎣a·COS(t) , b·SIN(t) ⎦
• Parametrization of a cardioid
o Syntax: Cardioid(a)
o Example: Cardioid(3/4)
⎛ 3 ⎞
#23: Cardioid⎜⎯⎟
⎝ 4 ⎠
to parametize the cardioid (x^2+y^2- 2 * 3/4x)^2 = 4 * (3/4)^2(x^2+y^2)
⎡ 2 ⎤
⎢ 3·COS(t) 3·COS(t) 3·SIN(t)·COS(t) 3·SIN(t) ⎥
#24: ⎢⎯⎯⎯⎯⎯⎯⎯⎯⎯ + ⎯⎯⎯⎯⎯⎯⎯⎯, ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ + ⎯⎯⎯⎯⎯⎯⎯⎯⎥
⎣ 2 2 2 2 ⎦
#25: Cardioid(a)

⎡ 2 ⎤
#26: ⎣2·a·COS(t) + 2·a·COS(t), 2·a·SIN(t)·COS(t) + 2·a·SIN(t)⎦
• Parametrization of a catenary
#27: Catenary(1)
o Syntax: Catenary(a)
o Example: Catenary(1)
to parametize the catenary y = 1 * cosh(x/1)
⎡ t -t ⎤
⎢ e e ⎥
#28: ⎢t, ⎯⎯ + ⎯⎯⎯⎥
⎣ 2 2 ⎦
#29: Catenary(a)
⎡ t/a - t/a ⎤
⎢ a·e a·e ⎥
#30: ⎢t, ⎯⎯⎯⎯⎯⎯ + ⎯⎯⎯⎯⎯⎯⎯⎯⎥
⎣ 2 2 ⎦

• Parametrization of a cicloid
o Syntax: Cicloid(r,h)
o Example: Cicloid(1,1)
#31: Cicloid(1, 1)
to parametize the cicloid [t - sin(t) , 1 - cos(t)]
#32: [t - SIN(t), 1 - COS(t)]
#33: Cicloid(r, h)
#34: [r·t - h·SIN(t), r - h·COS(t)]
• Parametrization of a cisoid
⎛ 3 ⎞
#35: Cisoid⎜⎯⎟
⎝ 4 ⎠
o Syntax: Cisoid(a)
o Example: Cisoid(3/4)
to parametize the cisoid x^3+xy^2 = 2 * 3/4 * y^2
⎡ 2 3 ⎤
⎢ 3·SIN(t) 3·SIN(t) ⎥
#36: ⎢⎯⎯⎯⎯⎯⎯⎯⎯⎯, ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎥
⎣ 2 2·COS(t) ⎦
#37: Cisoid(a)

⎡ 3 ⎤
⎢ 2 2·a·SIN(t) ⎥
#38: ⎢2·a·SIN(t) , ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎥
⎣ COS(t) ⎦
• Parametrization of Descartes' folium
o Syntax: DescartesFolium(a)
o Example: DescartesFolium(1.5)
#39: DescartesFolium(1.5)
to parametize the Descartes' folium x^3+y^3 = 3 * 1.5 * xy
⎡ 2 ⎤
⎢ 9·t 9·t ⎥
#40: ⎢⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯, ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎥
⎢ 3 3 ⎥
⎣ 2·(t + 1) 2·(t + 1) ⎦
#41: DescartesFolium(a)
⎡ 2 ⎤
⎢ 3·a·t 3·a·t ⎥
#42: ⎢⎯⎯⎯⎯⎯⎯, ⎯⎯⎯⎯⎯⎯⎥
⎢ 3 3 ⎥
⎣ t + 1 t + 1 ⎦

• Parametrization of EightFigure
o Syntax: EightFigure(a)
o Example: EightFigure(1.5)
#43: EightFigure(1.5)
to parametize the eight figure x^4 = 1.5^2(x^2-y^2)
⎡ 3·√(COS(2·t)) 3·SIN(t)·√(COS(2·t)) ⎤
⎢⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯, ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎥
#44: ⎢ 2·COS(t) 2 ⎥
⎣ 2·COS(t) ⎦
#45: EightFigure(a)
⎡ a·√(COS(2·t)) a·SIN(t)·√(COS(2·t)) ⎤
⎢⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯, ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎥
#46: ⎢ COS(t) 2 ⎥
⎣ COS(t) ⎦
• Parametrization of Pascal's Snail
o Syntax: PascalSnail(a,b)
o Example: PascalSnail(1,3/2)
⎛ 3 ⎞
#47: PascalSnail⎜1, ⎯⎟
⎝ 2 ⎠
to parametize the Pascal's snail (x^2+y^2-2*1*x)^2 = (3/2)^2(x^2+y^2)
⎡ 2 3·COS(t) 3·SIN(t) ⎤
#48: ⎢2·COS(t) + ⎯⎯⎯⎯⎯⎯⎯⎯, 2·SIN(t)·COS(t) + ⎯⎯⎯⎯⎯⎯⎯⎯⎥
⎣ 2 2 ⎦

#49: PascalSnail(a, b)
⎡ 2 ⎤
#50: ⎣2·a·COS(t) + b·COS(t), 2·a·SIN(t)·COS(t) + b·SIN(t)⎦
• Parametrization of Rosacea
o Syntax: Rosacea(a,k)
o Example: Rosacea(3,5)
#51: Rosacea(3, 5)
to parametize the rosacea [3 sin(5t)cos(t), 3 sin(5t)sin(t)]
#52: [3·COS(t)·SIN(5·t), 3·SIN(t)·SIN(5·t)]
#53: Rosacea(a, k)
#54: [a·COS(t)·SIN(k·t), a·SIN(t)·SIN(k·t)]

• Parametrization of Folium
o Syntax: Folium(a,b)
o Example: Folium(1,1)
#55: Folium(1, 1)
to parametize the folium (x^2+y^2) (x^2+y^2+x*1) = 4*1*xy^2
⎡ 2 2 3 ⎤
#56: ⎣COS(t) ·(4·SIN(t) - 1), COS(t)·(4·SIN(t) - SIN(t))⎦
#57: Folium(a, b)
⎡ 2 2 3 ⎤
#58: ⎣COS(t) ·(4·a·SIN(t) - b), COS(t)·(4·a·SIN(t) - b·SIN(t))⎦
• Parametrization of Hipocicloid
o Syntax: Hipocicloid(a,b)
o Example: Hipocicloid(1,1)
⎛ 3 ⎞
#59: Hipocicloid⎜2, ⎯⎟
⎝ 4 ⎠
to parametize the hipocicloid [(a-b)cos(t)+bcos((a-b)t/b),(a-b)sin(t)-bsin((a-b)t/b)]
⎡ ⎛ 5·t ⎞ ⎛ 5·t ⎞ ⎤
⎢ 3·COS⎜⎯⎯⎯⎟ 3·SIN⎜⎯⎯⎯⎟ ⎥
#60: ⎢ ⎝ 3 ⎠ 5·COS(t) 5·SIN(t) ⎝ 3 ⎠ ⎥
⎢⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ + ⎯⎯⎯⎯⎯⎯⎯⎯, ⎯⎯⎯⎯⎯⎯⎯⎯ - ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎥
⎣ 4 4 4 4 ⎦
#61: Hipocicloid(a, b)

⎡ ⎛ a·t ⎞ ⎛ a·t
#62: ⎢b·COS⎜⎯⎯⎯ - t⎟ + (a - b)·COS(t), (a - b)·SIN(t) - b·SIN⎜⎯⎯⎯ -
⎣ ⎝ b ⎠ ⎝ b
⎞⎤
t⎟⎥
⎠⎦
• Parametrization of Trisectriz
o Syntax: Trisectriz(a)
o Example: Trisectriz(1)
#63: Trisectriz(1)
to parametize the trisectriz y^2(1+x) = x^2(3*1-x)
⎡ 2 ⎤
#64: ⎣4·COS(t) - 1, 2·SIN(2·t) - TAN(t)⎦
#65: Trisectriz(a)
⎡ 2 ⎤
#66: ⎣4·a·COS(t) - a, 2·a·SIN(2·t) - a·TAN(t)⎦

• Parametrization of Tractriz
#67: Tractriz(3)
o Syntax: Tractriz(a)
o Example: Tractriz(3)
to parametize the tractriz [3 sin(t), 3 (cos(t) + ln(tan(t/2)))]
⎡ ⎛ ⎛ t ⎞⎞ ⎤
#68: ⎢3·SIN(t), 3·LN⎜TAN⎜⎯⎟⎟ + 3·COS(t)⎥
⎣ ⎝ ⎝ 2 ⎠⎠ ⎦
#69: Tractriz(a)
⎡ ⎛ ⎛ t ⎞⎞ ⎤
#70: ⎢a·SIN(t), a·LN⎜TAN⎜⎯⎟⎟ + a·COS(t)⎥
⎣ ⎝ ⎝ 2 ⎠⎠ ⎦
• Parametrization of Archimedes' spiral
o Syntax: ArchimedesSpiral(a)
o Example: ArchimedesSpiral(0.03)
#71: ArchimedesSpiral(0.03)
to parametize the Archimedes' spiral [at cos(t), at sin(t)]
⎡ 3·t·COS(t) 3·t·SIN(t) ⎤
#72: ⎢⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯, ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎥
⎣ 100 100 ⎦
#73: ArchimedesSpiral(a)
#74: [a·t·COS(t), a·t·SIN(t)]

• Parametrization of Astroid in R^3
o Syntax: Astroid3D(a,n)
o Example: Astroid3D(1,1,1)
#75: [COS(t), SIN(t), COS(2·t)]
to parametize the astroid [cos(t), sin(t), cos(2t)]
#76: Astroid3D(a, b, n)
⎡ n n ⎤
#77: ⎣a·COS(t) , b·SIN(t) , COS(2·t)⎦
#78: Astroid3D(1, 1, 1)
• Parametrization of spherical spiral
sin(1*t)]
o Syntax: SphericalSpiral(a,n,m)
o Example: SphericalSpiral(5,1,25)
to parametize the spherical spiral [5 cos(1*t)cos(25*t), 5 cos(1*t)sin(25*t), 5

#79: SphericalSpiral(5, 1, 25)
#80: [5·COS(t)·COS(25·t), 5·COS(t)·SIN(25·t), 5·SIN(t)]
#81: SphericalSpiral(a, n, m)
#82: [a·COS(m·t)·COS(n·t), a·SIN(m·t)·COS(n·t), a·SIN(n·t)]
• Parametrization of Viviani's curve
o Syntax: VivianiCurve(a)
o Example: VivianiCurve(2)
#83: VivianiCurve(2)
to parametize the Viviani's curve [2(1+cos(t)), 2 sin(t), 4 sin(t/2)]
⎡ ⎛ t ⎞⎤
#84: ⎢2·COS(t) + 2, 2·SIN(t), 4·SIN⎜⎯⎟⎥
⎣ ⎝ 2 ⎠⎦
#85: VivianiCurve(a)
⎡ ⎛ t ⎞⎤
#86: ⎢a·COS(t) + a, a·SIN(t), 2·a·SIN⎜⎯⎟⎥
⎣ ⎝ 2 ⎠⎦

• Parametrization of helicoidal curve
o Syntax: HelicoidalCurve(curve,c)
o Example: HelicoidalCurve(Circumference(0,0,r))
to parametize the helicoidal curve [r cos(t), r sin(t), ct]
#87: HelicoidalCurve(Circumference(0, 0, r))
#88: [r·COS(t), r·SIN(t), c·t]
#89: HelicoidalCurve(ArchimedesSpiral(a))
#90: [a·t·COS(t), a·t·SIN(t), c·t]
#91: HelicoidalCurve(Astroid(a))
⎡ 3 3 ⎤
#92: ⎣a·COS(t) , a·SIN(t) , c·t⎦

Exercise 1
Exercises
Compute the line integral of [xy^2+x+1, x^2y-2] along curve C given by:
•The ellipse (x-x_0)^2/a^2+(y-y_0)^2/b^2 = 1.
•The portion of the curve given by y=(x+2)\ln x + (1-x)^2 which joins the
point (1,0) with (-2,9).
⎡ 2 2 ⎤
#93: ExactDifferential(⎣x·y + x + 1, x ·y - 2⎦)
#94: This is an exact differential
⎡ 2 2 ⎤
#95: U(x, y) ≔ Potential(⎣x·y + x + 1, x ·y - 2⎦)
⎛ 2 ⎞
2 ⎜ y 1 ⎟
#96: U(x, y) ≔ x ·⎜⎯⎯ + ⎯⎟ + x - 2·y
⎝ 2 2 ⎠
#97: U(-2, 9) - U(1, 0)
285
#98: ⎯⎯⎯
2
Exercise 2
Compute the line integral of [e^x+1,x+z,xy+x+y+2 e^z] along the segment
which joins (0,1,2) with (2,-1,6).
⎡ x z⎤
#99: ExactDifferential(⎣e + 1, x + z, x·y + x + y + 2·e ⎦)
#100: This is not an exact differential
⎡ x z⎤
#101: LineIntegral(⎣e + 1, x + z, x·y + x + y + 2·e ⎦, Segment([0, 1,
2], [2, -1, 6]), 0, 1)
6 2 19
#102: 2·e - e - ⎯⎯
3

Exercise 3
Compute the area inside the cardiod (x^2+y^2- 2 * 3/4x)^2 = 4 * (3/4)^2(x^2+y^2).
Compute its length.
⎛ ⎛ 3 ⎞ ⎞
#103: AreaInsideCurve⎜Cardioid⎜⎯⎟, 0, 2·π⎟
⎝ ⎝ 4 ⎠ ⎠
27·π
#104: ⎯⎯⎯⎯
8
⎛ ⎛ 3 ⎞ ⎞
#105: CurveLength⎜Cardioid⎜⎯⎟, 0, 2·π⎟
⎝ ⎝ 4 ⎠ ⎠
#106: 12
Exercise 4
Let H be a wire which shape is the curve (cost,sint,t) ; t in [0,2pi] with
density function
ρ(x,y,z)=x^2+y^2+z^2. Compute the length, mass, mass center and media density
of the wire.
#107: CurveLength([COS(t), SIN(t), t], 0, 2·π)
#108: 2·√2·π
2 2 2
#109: WireMass(x + y + z , [COS(t), SIN(t), t], 0, 2·π)
3
8·√2·π
#110: ⎯⎯⎯⎯⎯⎯⎯ + 2·√2·π
3
2 2 2
#111: WireMassCenter(x + y + z , [COS(t), SIN(t), t], 0, 2·π)
⎡ 2 ⎤
⎢ 6 6·π 3·π·(2·π + 1) ⎥
#112: ⎢⎯⎯⎯⎯⎯⎯⎯⎯, - ⎯⎯⎯⎯⎯⎯⎯⎯, ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎥
⎢ 2 2 2 ⎥
⎣ 4·π + 3 4·π + 3 4·π + 3 ⎦
2 2 2
#113: MediumValue(x + y + z , [COS(t), SIN(t), t], 0, 2·π)
2
4·π + 3
#114: ⎯⎯⎯⎯⎯⎯⎯⎯
3