HP 33S Integral Demonstration and Programs
Integration
I'm going to start with a demonstration on how to calculate integrals with the HP 33S through the equation list. The steps will be similar for the HP 32SII and the later HP 35S.
1. Enter the lower limit, press [ ENTER ], the higher limit.
2. Press [ |→ ] [ STO ] (EQN), enter or select an equation.
3. Press [ |→ ] [ e^x ] ( ∫ ), to calculate the integration. You will be prompted for the variable to integrate.
Example: Fresnel Integral
Select Radians angle mode.
Fresnel Sine:
Equation: SIN(SQ(T)), variable to integrate: T
Note: [ x^2 ] -> SQ (square)
Use lower limit as 0.
S(2.4): lower = 0, upper = 2.4, S(2.4) ≈ 0.457864
S(5.8): lower = 0, upper = 5.8, S(5.8) ≈ 0.678004
Fresnel Cosine:
Equation: COS(SQ(T)), variable to integrate: T
Use lower limit as 0.
C(2.4): lower = 0, upper = 2.4, C(2.4) ≈ 0.510157
C(5.8): lower = 0, upper = 5.8, C(5.8) ≈ 0.695845
Source:
"Fresnel Integral" Wikipedia https://en.wikipedia.org/wiki/Fresnel_integral Retrieved May 2, 2022
Atmospheric Parameters
Given the height (H) in meters, the following can be estimated:
Temperature in Celsius (°C):
T ≈ 15.04 - 0.00649 * H
Pressure in kilopascals (kPa):
P ≈ 101.29 * ((T + 273.15) / 288.08)^5.256
(note, the source had 273.1 but I had 27.3.15 for better accuracy)
Atmospheric Density (km/m^3):
D ≈ P/(0.2869 * (T + 273.15))
Program:
HP 33S: Size: LN = 174, CK = 7625
A0001 LBL A
A0002 15.04
A0003 0.00649
A0004 INPUT H
A0005 ×
A0006 -
A0007 STO T
A0008 VIEW T
A0009 273.15
A0010 +
A0011 288.08
A0012 ÷
A0013 52.56
A0014 y^x
A0015 101.29
A0016 ×
A0017 STO P
A0018 VIEW P
A0019 0.2869
A0020 273.15
A0021 RCL+ T
A0022 ×
A0023 ÷
A0024 STO D
A0025 VIEW D
A0026 RTN
Example:
H = 50 m
Results:
T ≈ 14.715580 °C (about 58.4879 °F)
P ≈ 100.894225 kPa
D ≈ 1.221648 kg/m^3
Source:
"Engineering Formula Sheet" Project Lead The Way. https://www.madison-lake.k12.oh.us/userfiles/680/Classes/16192/IED-Review%20Engineering%20Formula%20Sheet.pdf Last Retrieved April 29, 2022
Legendre Polynomials
The value of Legendre Polynomials can be calculated using a closed formula from Rodrigues' formula:
P_n(x) = Σ( comb(n, k) * comb(n+k, k) * ((x - 1)/2)^k, k= 0, n)
Program:
HP 33S:
LBL L: Size: LN = 30, CK = 14EC
LBL M: Size: LN = 105, CK = AA05
Run XEQ L.
L0001 LBL L
L0002 0
L0003 STO P
L0004 INPUT X
L0005 INPUT N
L0006 STO K
M0001 LBL M
M0002 RCL N
M0003 RCL K
M0004 nCr
M0005 RCL N
M0006 RCL+ K
M0007 LASTx
M0008 nCr
M0009 ×
M0010 -1
M0011 RCL+ X
M0012 2
M0013 ÷
M0014 RCL K
M0015 y^x
M0016 ×
M0017 STO+ P
M0018 DSE K
M0019 GTO M
M0020 1
M0021 STO+ P
M0022 RCL P
M0023 RTN
Examples:
N = 2, X = 0.25; Result: -0.406250
N = 3, X = -0.46; Result: 0.446660
N = 4, X = 0.73; Result: -0.380952
Source:
"Legendre polynomials" Wikipedia. https://en.wikipedia.org/wiki/Legendre_polynomials Updated April 6, 2022. Last Accessed April 29, 2022.
Eddie
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