Line Integrals
Description
Published by Marcel Pelletier, July 2023, in the MoHPC - General Software Library
This document shows how to use the integral function ∫yx, in a program, to calculate the line integrals ∫c f(x,y) ds
Line integrals
Program D (Numerical Derivative) uses registers .0 , .1 and register I for indirect branching. Program E (Line integrals) use registers 1, 2, 3 and register I for indirect branching. The user-defined functions (label A, label B and label C) can be added at the end of the program memory. The formula used for the Line integrals is
∫c f(x,y) ds = ∫ab f(x(t), y(t)) √((dx/dt)2 + (dy/dt)2) dt
The function f(x,y)
is input on a parametric form f(x(t),y(t))
by the user. x(t)
at label A and y(t)
at label B.
The function f(x,y)
is input at label C, by replacing x by GSB A and y
by GSB B.
Examples
Example 1:
Evaluate ∫c(2+x²y) ds
where c is the upper half of the unit circle x² + y² = 1
. The parametric equations to represent c is x=cos
t and y=sin
t.
In PRGM mode:
Delete any existing program at and after label A.
f LBL A
COS
g RTN
f LBL B
SIN
g RTN
f LBL C
STO 0
GSB A
g x²
RCL 0
GSB B
×
2
+
g RTN
In RUN mode:
0 ENTER → 0.0000
g π f ∫xy E → 6.9499 (Value of the integral.)
The exact value is 2π+2/3
.
Example 2:
Evaluate ∫cxy⁴ds
where c is the right half of the unit circle x² + y² = 16
. The parametric equations to represent c is x=4cos
t and y=4sin
t and -π/2 ≤ t ≤ π/2
.
In PRGM mode:
Delete any existing program at and after label A.
f LBL A
COS
4
×
g RTN
f LBL B
SIN
4
×
g RTN
f LBL C
STO 0
GSB A
RCL 0
GSB B
4
yx
×
g RTN
In RUN mode:
g π CHS 2 ÷ → -1.5708
ENTER CHS f ∫xy E → 1,638.4000 (Value of the integral)
Program Resources
Labels
Name |
Description |
|
C |
# User function |
|
D |
Numerical Derivative |
|
E |
Line Integral |
|
Storage Registers
Name |
Description |
|
1 |
|
|
2 |
|
|
3 |
|
|
.0 |
Save x value |
|
.1 |
0.0001 |
|
I |
used for indirect branching |
|
Program
Line |
Display |
Key Sequence |
|
Line |
Display |
Key Sequence |
|
000 |
|
|
|
018 |
44 1 |
STO 1 |
|
001 |
42,21,14 |
f LBL D |
|
019 |
32 13 |
GSB C |
|
002 |
44 .0 |
STO . 0 |
|
020 |
44 2 |
STO 2 |
|
003 |
26 |
EEX |
|
021 |
2 |
2 |
|
004 |
16 |
CHS |
|
022 |
0 |
0 |
|
005 |
4 |
4 |
|
023 |
44 25 |
STO I |
|
006 |
44 .1 |
STO . 1 |
|
024 |
45 1 |
RCL 1 |
|
007 |
40 |
+ |
|
025 |
32 14 |
GSB D |
|
008 |
32 25 |
GSB I |
|
026 |
44 3 |
STO 3 |
|
009 |
45 .0 |
RCL . 0 |
|
027 |
2 |
2 |
|
010 |
45,30, .1 |
RCL − . 1 |
|
028 |
1 |
1 |
|
011 |
32 25 |
GSB I |
|
029 |
44 25 |
STO I |
|
012 |
30 |
− |
|
030 |
45 1 |
RCL 1 |
|
013 |
2 |
2 |
|
031 |
32 14 |
GSB D |
|
014 |
45,20, .1 |
RCL × . 1 |
|
032 |
45 3 |
RCL 3 |
|
015 |
10 |
÷ |
|
033 |
43 1 |
g →P |
|
016 |
43 32 |
g RTN |
|
034 |
45,20, 2 |
RCL × 2 |
|
017 |
42,21,15 |
f LBL E |
|
035 |
43 32 |
g RTN |
|