CURVELENGTH

Arc Length of a Parametric Curve for the HP-41

Overview

1°) 3-Dimensional Problem

   a) With Newton-Cotes Integration
   b) Discrete Data
   c) With Romberg's Method

2°) N-Dimensional Problem

a) With Newton-Cotes Integration
b) With Romberg's Method ( N < 10 )

Latest Update:

-The program in paragraph 1-a) has been improved to take into account the cylindrical & spherical coordinates.
-In paragraph 1-b) "CRVL" may be used if we only know a finite number of data points, equally spaced.

1°) 3-Dimensional Problem

a) With Newton-Cotes Integration

"CRVL" evaluates the integral §_a^b [ ( dx/dt )² + ( dy/dt )² + ( dz/dt )² ]^1/2 dt      with cartesian coordinates    ( CF 02-CF03 )
                    or §_a^b [ ( dr/dt )² + r² ( dL/dt )² + ( dz/dt )² ]^1/2 dt                         with cylindrical coordinates ( SF 02-CF 03 )
                    or §_a^b [ ( dr/dt )² + r² ( Cos b )² ( dL/dt )² + r² ( db/dt )² ]^1/2 dt     with spherical coordinates    ( CF 02-SF 03 )

-The derivatives are estimated by the following formula of order 10

df/dx ~ (1/2520.h).[ 2100.( f₁ - f_-1 ) - 600.( f₂ - f_-2 ) + 150.( f₃ - f_-3 ) - 25.( f₄ - f_-4 ) + 2.( f₅ - f_-5 ) ]

( f(x+k.h) is denoted f_k to simplify these expressions )

-And the integral is approximated by Newton-Cotes 7-point formula:

§_x1^x7 f(x).dx ~ h [ 41f(x₁)+216f(x₂)+27f(x₃)+272f(x₄)+27f(x₅)+216f(x₆)+41f(x₇) ] / 140 with x_i+1 - x_i = h = Cste

-The intervall [a,b] may be divided into n parts to get more and more accurate results.

Data Registers: • R00 = function name ( Registers R00 thru R03 are to be initialized before executing "CRVL" )

                                     • R01 = a      R04 = L             R06 to R19: temp
                                     • R02 = b      R05 = h   ( to estimate the derivatives )
                                     • R03 = n

Flags: F02-F03             CF 02 CF 03 ->   Cartesian coordinates
                                       SF 02   CF 03 ->   Cylindrical coordinates
                                       CF 02   SF 03 ->   Spherical coordinates

Subroutine: 1 program that takes t in X-register and returns x(t) , y(t) , z(t) in registers X , Y , Z respectively
or r(t) , L(t) , z(t) ---------------------------------
or r(t) , L(t) , b(t) ---------------------------------

-Line 28 is a three-byte GTO 01

01 LBL "CRVL"
02 STO 05
03 RCL 02
04 RCL 01
05 -
06 RCL 03
07 6
08 *
09 STO 09
10 /
11 STO 06
12 2
13 STO 10
14 STO 19
15 CLX
16 STO 04
17 600
18 CHS
19 STO 16
20 150
21 STO 17
22 14
23 *
24 STO 15
25 25
26 CHS
27 STO 18
28 GTO 01
29 LBL 00
30 RCL 06
31 RCL 09
32 *
33 RCL 01

34 +
35 STO 11
36 XEQ 02
37 CLX
38 SIGN
39 FC? 02
40 FS? 03
41 X=0?
42 GTO 00
43 D-R
44 FC? 43
45 ST* 13
46 SIGN
47 FS? 03
48 D-R
49 FC? 43
50 ST* 14
51 RCL 11
52 XEQ IND 00
53 ST* 13
54 FS? 03
55 ST* 14
56 X<> Z
57 COS
58 FS? 03
59 ST* 13
60 LBL 00
61 RCL 12
62 X^2
63 RCL 13
64 X^2
65 RCL 14
66 X^2

67 +
68 +
69 SQRT
70 RTN
71 LBL 02
72 5
73 STO 07
74 19
75 STO 08
76 CLX
77 STO 12
78 STO 13
79 STO 14
80 LBL 03
81 RCL 11
82 RCL 05
83 RCL 07
84 *
85 +
86 XEQ IND 00
87 RCL IND 08
88 ST* T
89 ST* Z
90 *
91 ST+ 12
92 RDN
93 ST+ 13
94 X<>Y
95 ST+ 14
96 RCL 11
97 RCL 05
98 RCL 07
99 *

100 -
101 XEQ IND 00
102 RCL IND 08
103 ST* T
104 ST* Z
105 *
106 ST- 12
107 RDN
108 ST- 13
109 X<>Y
110 ST- 14
111 DSE 08
112 DSE 07
113 GTO 03
114 RTN
115 LBL 01
116 XEQ 00
117 82
118 RCL 10
119 /
120 *
121 ST+ 04
122 CLX
123 SIGN
124 ST- 09
125 STO 10
126 XEQ 00
127 216
128 *
129 ST+ 04
130 DSE 09
131 XEQ 00
132 27

133 *
134 ST+ 04
135 DSE 09
136 XEQ 00
137 272
138 *
139 ST+ 04
140 DSE 09
141 XEQ 00
142 27
143 *
144 ST+ 04
145 DSE 09
146 XEQ 00
147 216
148 *
149 ST+ 04
150 DSE 09
151 GTO 01
152 XEQ 00
153 41
154 *
155 RCL 04
156 +
157 RCL 06
158 *
159 RCL 05
160 /
161 352800
162 /
163 STO 04
164 END

( 262 bytes / SIZE 020 )

STACK	INPUT	OUTPUT
X	h	L

Example1: x(t) = t² , y(t) = exp(t) , z(t) = Ln t ; a = 1 , b = 2

01 LBL "T"
02 X^2
03 LASTX
04 E^X
05 LASTX
06 LN
07 RTN

"T" ASTO 00 1 STO 01 2 STO 02 CF 02 CF 03

-If you choose n = 1 & h = 0.1 1 STO 03

0.1 XEQ "CRVL" >>>> L = 5.609133827 ---Execution time = 95s---

-With n = 2 - and the same h-value - it yields L = 5.609132375

-The exact result is L = 5.609132359...

Example2: r(t) = exp(t) , L(t) = t , b(t) = t² ; a = 0 , b = 1 ( all angles in radians )

01 LBL "T"
02 X^2
03 LASTX
04 ENTER^
05 E^X
06 RTN

XEQ "RAD" SF 02 CF 03

0 STO 01 1 STO 02

-With n = 1 & h = 0.1 1 STO 03

0.1 XEQ "CRVL" >>>> L = 2.645584611 ---Execution time = 84s---

-With n = 2 & h = 0.1 we get L = 2.645588402

-The exact value is 2.645588439...

Example3: r(t) = exp(t) , L(t) = t² , b(t) = t³ ; a = 0 , b = 1 ( all angles in radians )

01 LBL "T"
02 ENTER^
03 X^2
04 ST* Y
05 LASTX
06 E^X
07 RTN

XEQ "RAD" CF 02 SF 03

0 STO 01 1 STO 02

-With n = 1 & h = 0.1 1 STO 03

0.1 XEQ "CRVL" >>>> L = 3.449957472 ---Execution time = 90s---

-With n = 2 & h = 0.1 we get L = 3.449386896

-The exact result is 3.449386684...

Notes:

-"CRVL" works in RAD mode & in DEG mode but not in GRAD mode

-The spherical coordinates that are used here are the radius "vector" r , the longitude L and the latitude b
-If you prefer the co-latitude, replace line 57 COS by SIN

b) Discrete Data

-The following variant calculates the same integrals with the Simpson's rule and the derivatives with approximate formulas of order 4.

-The coordinates of the N points are to be stored in Rbb thru Ree ( bb > 15 )

Warning: the number of points N must be odd and at least 5

Data Registers: R00: temp ( Registers Rbb thru Ree are to be initialized before executing "CRVL" )

R02 to R15: temp

                                     • Rbb   = x₁         r₁          r₁ .................................. • Re-2 = x_n        r_n          r_n
                                     • Rb+1 = y₁   or L₁   or   L₁ .................................. • Re-1 = y_n   or L_n   or   L_n       bbb > 015
                                     • Rb+2 = z₁        z₁          b₁   .................................   • Ree   = z_n        z_n          b_n

Subroutines: /

01 LBL "CRVL"
02 STO 02
03 STO 03
04 INT
05 DSE X
06 CHS
07 LASTX
08 FRC
09 E3
10 *
11 STO 01
12 STO 10
13 +
14 3
15 STO 15
16 /
17 4
18 STO 13
19 -
20 STO 00
21 9
22 STO 14
23 CLX
24 STO 04
25 STO 05
26 STO 06
27 STO 07
28 STO 08
29 STO 09
30 SIGN
31 STO 11
32 STO 12
33 FC? 02
34 FS? 03
35 D-R

36 FC? 43
37 STO 11
38 SIGN
39 FS? 03
40 D-R
41 FC? 43
42 STO 12
43 25
44 XEQ 02
45 48
46 CHS
47 XEQ 02
48 36
49 XEQ 02
50 16
51 CHS
52 XEQ 02
53 XEQ 02
54 13
55 XEQ 03
56 X<> 01
57 STO 10
58 RCL 02
59 STO 03
60 CLX
61 STO 04
62 STO 05
63 STO 06
64 STO 07
65 STO 08
66 STO 09
67 3
68 XEQ 02
69 10
70 XEQ 02

71 18
72 CHS
73 XEQ 02
74 6
75 XEQ 02
76 SIGN
77 CHS
78 XEQ 02
79 10
80 XEQ 03
81 2
82 STO 09
83 X^2
84 STO 08
85 *
86 ST+ 01
87 GTO 01
88 LBL 02
89 SIGN
90 LBL 04
91 RCL IND 03
92 LASTX
93 *
94 ST- IND 13
95 RCL IND 10
96 LASTX
97 *
98 ST+ IND 14
99 1
100 ST+ 03
101 ST- 10
102 ST+ 13
103 ST- 14
104 DSE 15
105 GTO 04

106 3
107 ST- 13
108 ST+ 14
109 STO 15
110 RTN
111 LBL 03
112 ENTER
113 X<> 10
114 +
115 2
116 X<>Y
117 ST+ Y
118 RCL IND Y
119 COS
120 RCL IND Y
121 FS? 02
122 ST* 08
123 FS? 03
124 ST* 09
125 *
126 FS? 03
127 ST* 08
128 2
129 RCL 03
130 RCL 10
131 2
132 +
133 -
134 ST+ Y
135 RCL IND Y
136 COS
137 RCL IND Y
138 FS? 02
139 ST* 05
140 FS? 03

141 ST* 06
142 *
143 FS? 03
144 ST* 05
145 RCL 04
146 X^2
147 RCL 05
148 RCL 11
149 *
150 X^2
151 RCL 06
152 RCL 12
153 *
154 X^2
155 +
156 +
157 SQRT
158 RCL 07
159 X^2
160 RCL 08
161 RCL 11
162 *
163 X^2
164 +
165 RCL 09
166 RCL 12
167 *
168 X^2
169 +
170 SQRT
171 +
172 RTN
173 LBL 05
174 SIGN
175 LBL 06

176 RCL IND 02
177 LASTX
178 *
179 ST+ IND 13
180 1
181 ST+ 02
182 ST+ 13
183 DSE 15
184 GTO 06
185 3
186 ST- 13
187 STO 15
188 RTN
189 LBL 01
190 CLX
191 STO 04
192 STO 05
193 STO 06
194 SIGN
195 XEQ 05
196 8
197 CHS
198 XEQ 05
199 ST+ 02
200 8
201 XEQ 05
202 SIGN
203 CHS
204 XEQ 05
205 2
206 RCL 02
207 9
208 -
209 ST+ Y
210 RCL IND Y

211 COS
212 RCL IND Y
213 FS? 02
214 ST* 05
215 FS? 03
216 ST* 06
217 *
218 FS? 03
219 ST* 05
220 RCL 04
221 X^2
222 RCL 05
223 RCL 11
224 *
225 X^2
226 RCL 06
227 RCL 12
228 *
229 X^2
230 +
231 +
232 SQRT
233 RCL 08
234 X<> 09
235 STO 08
236 *
237 ST+ 01
238 12
239 ST- 02
240 DSE 00
241 GTO 01
242 36
243 ST/ 01
244 RCL 01
245 END

( 361 bytes / SIZE var )

STACK	INPUT	OUTPUT
X	bbb.eee	L

With bbb > 015

Example1: x(t) = t² , y(t) = exp(t) , z(t) = Ln t ; a = 1 , b = 2 but you only have the coordinates of 9 points corresponding to t = 1 , 9/8 , ... , 15/8 , 2

( you can store these coordinates with the following routine, for example in R21 to R47 )

LBL 10
9
STO 00
47
STO 01
LBL 01
RCL 00
1
-
8
/
1
+
LN
STO IND 01
DSE 01
LASTX
E^X
STO IND 01
E^X
DSE 01
LASTX
X^2
STO IND 01
DSE 01
DSE 00
GTO 01

CF 02 CF 03 21.047 XEQ "CRVL" >>>> L = 5.609125853 ---Execution time = 66s---

-The exact result is L = 5.609132359...

Example2: r(t) = exp(t) , L(t) = t , b(t) = t² ; a = 0 , b = 1 ( all angles in radians )

but you only have the coordinates of 9 points corresponding to t = 0 , 1/8 , ... , 7/8 , 1

( you can store these coordinates with the following routine, for example in R21 to R47 )

LBL 10
9
STO 00
47
STO 01
LBL 01
RCL 00
1
-
8
/
X^2
STO IND 01
DSE 01
LASTX
STO IND 01
DSE 01
E^X
STO IND 01
DSE 01
DSE 00
GTO 01

XEQ "RAD" SF 02 CF 03

21.047 XEQ "CRVL" >>>> L = 2.645598006 ---Execution time = 66s---

-The exact value is 2.645588439...

Example3: r(t) = exp(t) , L(t) = t² , b(t) = t³ ; a = 0 , b = 1 ( all angles in radians )

but you only have the coordinates of 9 points corresponding to t = 0 , 1/8 , ... , 7/8 , 1

( you can store these coordinates with the following routine, for example in R21 to R47 )

LBL 10
9
STO 00
47
STO 01
LBL 01
RCL 00
1
-
8
/
X^2
LASTX
*
STO IND 01
DSE 01
LASTX
X^2
STO IND 01
DSE 01
LASTX
E^X
STO IND 01
DSE 01
DSE 00
GTO 01

XEQ "RAD" CF 02 SF 03

21.047 XEQ "CRVL" >>>> L = 3.449756039 ---Execution time = 67s---

-The exact result is 3.449386684...

Notes:

-"CRVL" works in RAD mode & in DEG mode but not in GRAD mode
-This routine does not check if the control number is correct !

-The spherical coordinates that are used here are the radius "vector" r , the longitude L and the latitude b
-If you prefer the co-latitude, replace lines 119-136-211 by SIN

c) With Romberg Method

-This variant uses Pythagoreas theorem and Romberg method to evaluate the integral

Data Registers: • R00 = function name ( Register R00 is to be initialized before executing "CRVL" )

R01 = a R03 = L R04 to R16......: temp
R02 = b
Flags: /
Subroutine: 1 program that takes t in X-register and R07 and returns X(t) , Y(t) , Z(t) in registers X , Y , Z respectively

-Line 96 is a three-byte GTO 01

01 LBL "CRVL"
02 STO 02
03 X<>Y
04 STO 01
05 1
06 STO 04
07 LBL 01
08 RCL 02
09 RCL 01
10 -
11 RCL 04
12 STO 05
13 /
14 STO 11
15 CLX
16 STO 06
17 RCL 02
18 STO 07
19 XEQ IND 00
20 STO 08

21 RDN
22 STO 09
23 X<>Y
24 STO 10
25 LBL 02
26 RCL 05
27 RCL 11
28 ST* Y
29 -
30 RCL 01
31 +
32 STO 07
33 XEQ IND 00
34 ENTER
35 X<> 08
36 -
37 X^2
38 X<>Y
39 ENTER
40 X<> 09

41 -
42 X^2
43 +
44 X<>Y
45 ENTER
46 X<> 10
47 -
48 X^2
49 +
50 SQRT
51 ST+ 06
52 DSE 05
53 GTO 02
54 RCL 06
55 SIGN
56 ST* X
57 RCL 04
58 ST+ 04
59 X#Y?
60 GTO 00

61 STO 14
62 LASTX
63 STO 16
64 GTO 01
65 LBL 00
66 4
67 STO 12
68 16
69 STO 13
70 RCL 14
71 STO 15
72 LASTX
73 ISG 14
74 LBL 03
75 ENTER
76 ENTER
77 X<> IND 13
78 -
79 RCL 12
80 4

81 ST* 12
82 SIGN
83 ST+ 13
84 -
85 /
86 +
87 DSE 15
88 GTO 03
89 STO 03
90 STO IND 13
91 VIEW X
92 RND
93 X<>Y
94 RND
95 X#Y?
96 GTO 01
97 RCL 03
98 END

( 132 bytes / SIZE 016+??? )

STACK	INPUTS	OUTPUTS
Y	a	/
X	b	L(a,b)

Example: X(t) = t⁴ , Y(t) = t² , Z(t) = exp t ; a = 0 , b = 1

01 LBL "T"
02 E^X
03 LASTX
04 X^2
05 ENTER^
06 X^2
07 RTN

-If you try 9 decimals

T ASTO 00
FIX 9

0 ENTER^
1 XEQ "CRVL" >>>> L = 2.342116456 ---Execution time = 97s---

Notes:

-The HP41 displays the successive approximations.
-The result depends on the display format ( lines 92-94 )
-The exact result is L = 2.342116459....

2°) N-Dimensional Problem

a) With Newton-Cotes Integration

"CRVLN" evaluates the integral §_a^b [ ( dX₁/dt )² + ( dX₂/dt )² + ............ + ( dX_N/dt )² ]^1/2 dt

-The derivatives and the integral are estimated by the same formulas of order 10.

-The inervall [a,b] may be divided in m parts to get more and more accurate results.

Data Registers: R00: temp

                                        R01 = a      R04 = h      R06 to R15: temp
                                        R02 = b      R05 = L
                                        R03 = m
Flags: /
Subroutine: N programs named "X1" "X2" ............. "XN"   that take t in X-register and return X₁(t) , X₂(t) , .......... , X_N(t) respectively in X-register

01 LBL "CRVLN"
02 "DIM=?"
03 PROMPT
04 STO 14
05 "H^M^A^B"
06 PROMPT
07 STO 02
08 X<>Y
09 STO 01
10 -
11 X<>Y
12 STO 03
13 6
14 *
15 STO 07
16 /
17 STO 06
18 X<>Y
19 STO 04
20 6
21 *
22 STO 09
23 FIX 0
24 CF 29

25 XEQ 00
26 41
27 *
28 STO 05
29 DSE 07
30 GTO 01
31 LBL 00
32 RCL 06
33 RCL 07
34 *
35 RCL 01
36 +
37 STO 11
38 CLX
39 STO 13
40 RCL 14
41 STO 15
42 LBL 04
43 "X"
44 ARCL 15
45 XEQ 02
46 X^2
47 ST+ 13
48 DSE 15

49 GTO 04
50 RCL 13
51 SQRT
52 RTN
53 LBL 02
54 ASTO 00
55 RCL 09
56 STO 10
57 XEQ 03
58 ST+ X
59 STO 08
60 XEQ 03
61 25
62 *
63 ST- 08
64 XEQ 03
65 150
66 *
67 ST+ 08
68 XEQ 03
69 600
70 *
71 ST- 08
72 XEQ 03

73 2100
74 *
75 RCL 08
76 +
77 RCL 04
78 2520
79 *
80 /
81 RTN
82 LBL 03
83 RCL 04
84 ST- 10
85 RCL 11
86 RCL 10
87 -
88 XEQ IND 00
89 STO 12
90 RCL 11
91 RCL 10
92 +
93 XEQ IND 00
94 RCL 12
95 -
96 RTN

97 LBL 01
98 XEQ 00
99 216
100 *
101 ST+ 05
102 DSE 07
103 XEQ 00
104 27
105 *
106 ST+ 05
107 DSE 07
108 XEQ 00
109 272
110 *
111 ST+ 05
112 DSE 07
113 XEQ 00
114 27
115 *
116 ST+ 05
117 DSE 07
118 XEQ 00
119 216
120 *

121 ST+ 05
122 DSE 07
123 CLX
124 XEQ 00
125 STO M
126 82
127 *
128 ST+ 05
129 DSE 07
130 GTO 01
131 CLX
132 X<> M
133 41
134 *
135 ST- 05
136 RCL 05
137 RCL 06
138 *
139 140
140 /
141 STO 05
142 FIX 9
143 SF 29
144 END

( 251 bytes / SIZE 016 )

STACK	INPUT	OUTPUT
X	/	L(a,b)

Example: X₁(t) = t⁴ , X₂(t) = t² , X₃(t) = t³ , X₄(t) = exp t ; a = 0 , b = 1

01 LBL "X1"
02 X^2
03 X^2
04 RTN
05 LBL "X2"
06 X^2
07 RTN
08 LBL "X3"
09 ENTER^
10 X^2
11 *
12 RTN
13 LBL "X4"
14 E^X
15 RTN

-With h = 0.1 and m = 1

XEQ "CRVLN" >>>> "DIM=?"

4 R/S "H^M^A^B"

             0.1   ENTER^
               1    ENTER^
               0    ENTER^
               1        R/S          >>>>      L = 2.579075091                        ---Execution time = 180s---

-With m = 2 - and the same h-value - it yields L = 2.579079247

-The exact result is L = 2.579079231....

b) With Romberg Method ( n < 10 )

-Like the 2nd version of "CRVL", this variant of "CRVLN" uses an extrapolation to the limit.

Data Registers: R00 = n

                                        R01 = a      R03 = L     R04 to R26......: temp
                                        R02 = b
Flags: /
Subroutine: N programs named "X1" "X2" ....... "XN"   that take t in X-register & R07 and return X₁(t) , X₂(t) , ....... , X_N(t) respectively in X-register

-Line 115 is a three-byte GTO 01

01 LBL "CRVLN"
02 STO 02
03 RDN
04 STO 01
05 X<>Y
06 STO 00
07 1
08 STO 04
09 RCLFLAG
10 STO 20
11 LBL 01
12 FIX 00
13 CF 29
14 RCL 00
15 STO 18
16 17
17 STO 19
18 LBL 02
19 "X"
20 ARCL 18
21 ASTO L
22 RCL 02
23 STO 07
24 XEQ IND L

25 STO IND 19
26 DSE 19
27 DSE 18
28 GTO 02
29 CLX
30 STO 06
31 RCL 02
32 RCL 01
33 -
34 RCL 04
35 STO 05
36 /
37 STO 08
38 LBL 03
39 RCL 05
40 RCL 08
41 ST* Y
42 -
43 RCL 01
44 +
45 STO 07
46 RCL 00
47 STO 18
48 17

49 STO 19
50 CLX
51 STO 21
52 LBL 04
53 "X"
54 ARCL 18
55 ASTO L
56 RCL 07
57 XEQ IND L
58 ENTER
59 X<> IND 19
60 -
61 X^2
62 ST+ 21
63 DSE 19
64 DSE 18
65 GTO 04
66 RCL 21
67 SQRT
68 ST+ 06
69 DSE 05
70 GTO 03
71 RCL 20
72 STOFLAG

73 RCL 06
74 SIGN
75 ST* X
76 RCL 04
77 ST+ 04
78 X#Y?
79 GTO 00
80 STO 24
81 LASTX
82 STO 26
83 GTO 01
84 LBL 00
85 4
86 STO 22
87 26
88 STO 23
89 RCL 24
90 STO 25
91 LASTX
92 ISG 24
93 LBL 05
94 ENTER
95 ENTER
96 X<> IND 23

97 -
98 RCL 22
99 4
100 ST* 22
101 SIGN
102 ST+ 23
103 -
104 /
105 +
106 DSE 25
107 GTO 05
108 STO 03
109 STO IND 23
110 VIEW X
111 RND
112 X<>Y
113 RND
114 X#Y?
115 GTO 01
116 RCL 03
117 END

( 184 bytes / SIZE 026+??? )

STACK	INPUTS	OUTPUTS
Z	n < 10	/
Y	a	/
X	b	L(a,b)

Example: X₁(t) = t⁴ , X₂(t) = t² , X₃(t) = t³ , X₄(t) = exp t ; a = 0 , b = 1

01 LBL "X1"
02 X^2
03 X^2
04 RTN
05 LBL "X2"
06 X^2
07 RTN
08 LBL "X3"
09 ENTER^
10 X^2
11 *
12 RTN
13 LBL "X4"
14 E^X
15 RTN

-If you try to find 9 decimals

FIX 9

    4   ENTER^
    0   ENTER^
    1   XEQ "CRVLN" >>>>      L = 2.579079231                                ---Execution time = 4m16s---

Notes:

-The HP41 displays the successive approximations.
-The result depends on the display format ( lines 111-113 )
-The exact result is L = 2.579079231....

-So, there is no roundoff error in this example ( this is not always the case... ).

hp41programs

Arc Length of a Parametric Curve for the HP-41