RPN: Certain Integrals to Positive Infinity

RPN: Certain Integrals to Positive Infinity

Introduction

Today’s RPN session deals with improper integrals where the upper limit is positive infinity (∞).

∫( f(x) dx, a, ∞) → lim ∫( f(x) dx, a, t) as t → ∞

Some properties that will be use:

lim 1/t^n as t → ∞ approaches 0

lim e^(-t) as t → ∞ approaches 0

lim p(t) ÷ q(t) as t → ∞ approaches 0 where p(t) and q(t) are polynomials and

degree p(t) < degree q(t)

lim p(t) ÷ q(t) as t → ∞ approaches p_n ÷ q_n where p(t) and q(t) are polynomials and

degree p(t) = degree q(t). p_n and q_n are the leading coefficients of p(t) and q(t), respectively.

All integrals presented will have closed formulas because they have very friendly anti-derivatives.

∫( 1 ÷ (x^n) dx, a, ∞)

∫( 1 ÷ (x^n) dx, a, ∞) = 1 ÷ ((n - 1) * a^(n - 1))

(Abbreviated) Derivation:

∫( 1 ÷ (x^n) dx, a, ∞)

= lim (1 ÷ x^(n - 1) * -1 ÷ (n – 1) as x → ∞) - (1 ÷ a^(n – 1) * -1 ÷ (n – 1))

= ( 0 * -1 ÷ (n – 1) ) + 1 ÷ ((n – 1) * a^(n – 1))

= 1 ÷ ((n – 1) * a^(n – 1))

∫( 1 ÷ (x^n) dx, a, ∞) = 1 ÷ ((n - 1) * a^(n - 1))

HP 15C Code:

[42, 21, 11]: LBL A

[         1]: 1

[        30]: -

[    43, 36]: LSTx

[        20]: ×

[        15]: 1/x

[    43, 32]: RTN

Stack:

Y: a

X: n

Examples:

a = 2.75, n = 2: 4/11 ≈ 0.36364

a = 4.9, n = 3: ≈ 0.02082

∫( 1 ÷ ((x - r)*(x - s)) dx, a, ∞)

∫( 1 ÷ ((x - r)*(x - s)) dx, a, ∞)

= 1 ÷ (r - s) * ln( abs((a - s) ÷ (a – r)) )

For best results, a > max(s, r)

(Abbreviated) Derivation:

∫( 1 ÷ ((x - r)*(x - s)) dx, a, ∞)

Simply by partial fractions:

1 ÷ ((x - r)*(x – s)) = 1 ÷ ((r – s) * (x – r)) – 1 ÷ ((r – s) * (x – s))

Anti-derivative:

∫( 1 ÷ ((x - r)*(x - s)) dx)

= 1 ÷ (r – s) * ( ln(abs(x – r)) – ln(abs(x – s)) )

= 1 ÷ (r – s) * ln ( abs((x – r) ÷ (x – s)) )

Limit as x → ∞:

1 ÷ (r – s) * ln ( abs((x – r) ÷ (x – s)) )

= 1 ÷ (r – s) * ln ( abs((1 – r ÷ x) ÷ (1 – s ÷ x)) )

= 1 ÷ (r – s) * ln ( abs(1) )

= 1 ÷ (r – s) * ln(1)

= 0

When x = a

1 ÷ (r – s) * ln ( abs((a – r) ÷ (a – s)) )

= 1 ÷ (r – s) * ln ( abs(1 ÷ [(a – r) * (a -s)]) )

= 1 ÷ (r – s) * ln ( 1 ÷ [abs((a – s) ÷ (a – r))] )

= -1 ÷ (r – s) * ln ( abs((a – s) ÷ (a – r)) )

Then:

∫( 1 ÷ ((x - r)*(x - s)) dx, a, ∞) = 0 - (-1 ÷ (r – s) * ln ( abs((a – s) ÷ (a – r)) ))

= 1 ÷ (r - s) * ln ( abs((a - s) ÷ (a – r)) )

Code:

[42, 21, 12]: LBL B

[    44,  1]: STO 1

[        33]: R↓

[    44,  2]: STO 2

[        33]: R↓

[    44,  3]: STO 3

[45, 30,  1]: RCL- 1

[    45,  3]: RCL 3

[45, 30,  2]: RCL- 2

[        10]: ÷

[    43, 16]: ABS

[    43, 12]: LN

[    45,  2]: RCL 2

[45, 30,  1]: RCL- 1

[        15]: 1/x

[        20]: ×

[    43, 32]: RTN

Stack:

Z: a

Y: r

X: s

Examples:

a = 7.25, b = -3, s = 6: (ln 41 - ln 5) ÷ 9 ≈ 0.23379

a = 11, b = 4, s = 9: (ln 7 - ln 2) ÷ 5 ≈ 0.25055

∫( 1 ÷ e^x dx, a, ∞)

∫( 1 ÷ e^x dx, a, ∞) = 1 ÷ e^a

Code:

[42, 21, 13]: LBL C

[        12]: e^x

[        15]: 1/x

[    43, 32]: RTN

Stack:

X: a

Examples:

a = 4: e^(-4) ≈ 0.01832

a = 6: e^(-6) ≈ 0.00248

∫( x ÷ e^x dx, a, ∞)

∫( x ÷ e^x dx, a, ∞) = (a + 1) ÷ e^a

Code:

[42, 21, 14]: LBL D

[        12]: e^x

[    43, 36]: LSTx

[         1]: 1

[        40]: +

[        34]: x<>y

[        10]: ÷

[    43, 32]: RTN

Stack:

X: a

Examples:

a = 4: 5 ÷ e^(-4) ≈ 0.09158

a = 6: 7 ÷ e^(-6) ≈ 0.01735

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

The author does not use AI engines and never will.

Quick Review: Casio fx-9910CW 2nd Edition

Quick Review: Casio fx-9910CW 2^nd Edition

The Casio fx-9910CW 2^nd Edition is an update of the fx-991CW first released in 2023.

Product Pages

United States:

https://www.casio.com/us/scientific-calculators/product.FX-9910CW/

The fx-9910CW has two keyboards: black with gold font and pink with purple font (limited). I have one each and Casio has improved on the readability on the pink edition with the dark purple font.

I’m sure this model will be available world wide soon.

What’s the Same and What’s Different

Overall, the fx-9910CW 2^nd Edition has the same mathematical features as the fx-991CW. The modes included are:

Calculate: the main app for mathematical calculations

Statistics: 1 and 2 variable with seven regression models:

Linear:  y=a+bx

Quadratic: y=a+bx+cx^2

Logarithmic:  y=a+b*ln(x)

Exponential:  y=a*e^(bx)

Power I:  y=a*b^x

Power II: y=a*x^b

Inverse:  y=a+b/x

Statistical graphs may be generated with the QR feature.

Distribution: Binomial, Normal, and Poisson distributions, along with their inverses. The functions work with lower-tail probabilities (-∞ or 0 to x).

Spreadsheet: 5 columns, 45 rows. 2,380 byte memory.

Table: Generate a table for up to two functions f(x) and g(x). Generate graphs with the QR feature.

Equations: Linear systems (up to 4 x 4), polynomials up to 4^th order, and a general equation solver.

Inequalities: Solve inequalities for polynomials up to the 4^th order.

Complex Numbers: Complex number arithmetic with polar/rectangular conversion, integer powers, real/imaginary parts, conjugate

Base N: Base mode for the traditional decimal, hexadecimal, octal, and binary basis. Binary integers are up to 31 bytes with one sign bit.

Matrix: Four matrices, up to size 4 x 4, with basic matrix functions such as determinant, transpose, and inverse.

Vector: Four vectors, 2D or 3D, with dot product, cross product, and norm.

Math Box: Dice Roll and Coin Toss simulations

47 scientific constants (SI units) and unit conversions

We still have the newer Casio style of navigation keys, the Home Key, the Back Key, the Settings key, and the Scroll up and down keys.

What is different?

First, the fx-9910CW is not solar powered, but instead runs on a single AAA battery.

Welcome changes:

The menus all have short cut keys! This eliminates the requirement to scroll down menus, which some get long, to find the sub menus and functions. This makes the operating the calculator a lot easier and more efficient, eliminating additional key strokes by repeatedly pressing the arrow keys.

For example, to get the absolute value in Calculate mode, press [ CATALOG ], [ 3 ] for Numeric Calc, [ 1 ] for Absolute Value.

In the same mode, we call up the Speed of Light constant (c), press [ CATALOG ] , [ 7 ] for Sci Constants, [ 1 ] for Universal, and [ 3 ] for c.

There is a setting to turn the hide the shortcut numbers but thankfully it won’t turn off the ability to navigate by using short cut keys.

The 10^[] and FORMAT keys return to familiar form! Like the fx-CG 100, we have an option on how the 10^[] and FORMAT keys operate.

10^[] Key:

(1) Power: Acts like a power key like the fx-991CW. I read that there are potential problems with calculating with this key leaving calculations to return unexpected answers.

(2) Sci. Notat: This returns this key to the traditional scientific notation key. When this option is selected, the 10^ is shown in a small font. This key now acts like the [ EE ] or [ EXP ] keys on other scientific calculators. This is the default setting.

FORMAT Key:

(1) -π√ ←→ Decimal. Pressing the FORMAT key just toggles between decimal and the exact (when available) format of answers. This brings back the beloved [ S←→D ] key. This is the default settings.

(2) Format Menu. This is the format menu that is presented, similar to the fx-991CW. We trade off the quick toggle for additional formatting options.

Shortcut Catalog. Pressing [ SHIFT ] [ CATALOG ] is a new feature and provides a listing of all mathematical functions available in the active mode. Up to 15 functions are shown, using the arithmetic keys (+, -, ×, ÷) and the decimal point (.) as additional shortcut keys. The shortcut catalog does not have the conversions or constants.

These changes are welcome and enhance the operating experience of the calculator.

My next wish is that the fx-9910CW included the Algo mode (algorithm mode) that allowed small programs (via a mix of Scratch and Casio Basic). You can find out more details here:

https://edspi31415.blogspot.com/2025/08/casio-fx-92-college-plotting-lines.html

I just think the algorithm mode is neat, gives students an introduction to programming, and would have fit the fx-9910CW well.

Great job, Casio. I think Casio listened to calculator users and brought back these improvements, and brought it more in line with operating the fx-CG 100/Graph Math Plus. If you were hesitant about the fx-991CW, the fx-9910CW 2^nd Edition is a good time to jump in.

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

fx-3650P and TI-68: Quadratic Equation and Arc Length between Roots of a Quadratic Curve

fx-3650P and TI-68: Quadratic Equation and Arc Length between Roots of a Quadratic Curve

Two approaches using two well-liked, classic calculators. The fx-3650P uses Basic like language while the TI-68 handles formulas.

Quadratic Equation

Solve for A x^2 + B x + C = 0, with the discriminant D = B^2 – 4 * A * C.

We know the solutions: X = (-B ± √(B² – 4 * A * C)) / (2 * A)

fx-3650P Program

? → A : ? → B : ? → C :

B² – 4 A C → D ◢

-B ÷ ( 2 A ) → M :

√ ( √ ( D² ) ) ÷ ( 2 A ) → Y :

D ≥ 0 ⇒ Goto 1 : M ◢ Y ◢ Goto 2 :

Lbl 1 : M + Y → X ◢ M – Y → Y ◢ Lbl 2

D: discriminant

If D<0; roots are in the form of M ± Yi

Else, the roots are real and are stored in X, Y

TI-68 Formula

X = 0 × A + (-B + √(B² – 4 × C × A) × J) ÷ (2 × A)

The 0 × A is added to force A to be prompted for first.

J = -1 for one root, J = 1 for the other

TI-68 takes care of both real and complex roots, no worries.

The coefficients can be complex!

Examples

A	B	C	D	Roots
2	-3	-9	81	3, -1.5
1	0	25	-100	5i, -5i
-48	64	28	9472	-0.347127088, 1.680464022

Arc Length of a Quadratic Equation between its Real Roots

Give roots X, Y: (t – X) * (t – Y) = t^2 – (X + Y )* t + X * Y

f(t) = t^2 – (X + Y) * t + X * Y

f’(t) = 2 * t – (X + Y)

arc = ∫( √(1 + f’(t)^2) dt

TI-68 will set up for the outside integral function, while the fx-3650P can use the integral function inside of the program.

We are going to assume that X < Y.

fx-3650P Program

? → A : ? → B : ∫ ( √ (1 + (2 X – A – B) ² ), A, B)

This is the direct approach.

TI-68 Formula

ARC = √(1 + (2 × X – A – B)²)

for X use the integral function (dx)

[ 3^rd ] [ Σ+ ] (dx) [ = ]

Enter low, high, and the number of intervals.

The more intervals, generally, the more accurate the integral is.

Examples

I compared results against the fx-991CW.

A	B	TI-68, intv = 16	fx-3650P	fx-991 CW
2	9	26.070832160	26.070800000	26.070797720
0	5	13.903768900	13.904000000	13.903767950
-2	2	9.293567375	9.293568000	9.293567525

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

The author does not use AI engines and never will.

HP 67 Programs… Almost 50 Years Later

 HP 67 Programs… Almost 50 Years Later

Both downloads are in PDF format. This is for use for the HP 67 and its emulators, or really almost any RPN scientific calculator. Enjoy!

Volume 1:

https://drive.google.com/file/d/114H4D0hcOjxDj_MHQNvwDBUJV3chlpdC/view?usp=sharing

Countdown of HP 67 "seconds"

Random Numbers

Snell's Law

Circle: Area and Circumference

Sphere: Surface Area and Volume

Angle Between 2 Lines with Slopes x and y

Sum of Powers

Adding Complex Numbers

Multiplying Complex Numbers

Complex Number to a Real Power

Permutation

Combination (with duplicating X and Y stack values)

Speed of Sound Approximation (in meters per second)

Finance: Present Value Annuity Factor (including setting N and I% with monthly payments)

Distance Between Two Points (x,y) and (z,t)

Horizontal Curve

Volume 2:

https://drive.google.com/file/d/1RPK2C879JeiyReox1SOTeAOQc_7QYdUN/view?usp=sharing

Sigmoid Function

Logit Function

Solving Linear Equations

Solving Monic Quadratic Polynomials (real roots only)

Intensity of a Spherical Light Source

Determinant of a 2 x 2 Matrix

Air Density of Dry Air

Time Dilation Factor

Simple Interest

Calculus: Area Under the Curve of a Linear Function

Horizontal Curve Solution given Radius and Central Angle

Freezing Altitude Levels: Dry and Wet

HP 67 Solvers:

Temperature Conversions

Cost-Sell-Margin

Decibel Gain/Loss

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Casio fx-991CW: Arc of a Quadratic Curve

Casio fx-991CW: Arc of a Quadratic Curve

All screen shots will be taken with the https://www.classpad.app/ website.

Introduction

We are given three points: (x1, y1), (x2, y2), and (x3, y3) which are connected to a quadratic curve:

y(x) = a + b * x + c * x^2

with it’s derivative:

y’(x) = b + 2 * c * x

We assume that all three x-values are unique, therefore x1 ≠ x2, x2 ≠ x3, and x1 ≠ x3.

The arc length is the integral: ∫( √(1 + (b + 2 * c * x)^2 dx, min(x), max(x))

In curve fitting, if we only have three points, the quadratic curve will fit all three points.

Procedure

We can do the entire calculation in the Statistics app of the fx-991CW.

1. Go to the Statistics app by pressing [ Home ], selecting Statistics.

2. Enter the three points. If you need to, you can clear the lists by selecting Tools > Edit > Delete All.

3. Press [OK] (or [EXE] ), select Statistics Calc, y = a + b * x + c * x^2, and then press [OK] (or [EXE]) again. The Statistics Calc will allow us to make calculations using the statistics variables.

4. Calculate the integral:

∫ ( √(1 + (b + 2 * c * x)^2), min(x), max(x))

∫: Catalog > Func Analysis > Integration

b: Catalog > Statistics > Regression > b

c: Catalog > Statistics > Regression > c

min(x): Catalog > Statistics > Regression > Min/Max > min(x)

max(x): Catalog > Statistics > Regression > Min/Max > min(x)

Let’s walk through an example:

Points: (0,0), (5,4), (7,2)

Arc length: 9.97139451

You can see the procedure through the screen shots below:

Quadratic Curve:

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Numworks: Estimating the Speed of Sound in Water

 Numworks: Estimating the Speed of Sound in Water

Generally, the speed of sound in water is faster than the speed of sound in air. Is there an easy way to estimate the speed of sound in water using curve fitting? I will use the Numworks’ Regression App. The online simulator of Numworks: https://www.numworks.com/simulator/ .

* For readability, I consolidated some screen shots.

The data, from The Engineering Toolbox (see Source below) is shown in below:

SI Units

Temperature in °C	Speed of sound in m/s (meters/second)
0	1403
5	1427
10	1447
20	1481
30	1507
40	1526
50	1541
60	1552
70	1555
80	1555
90	1550
100	1543

https://www.engineeringtoolbox.com/sound-speed-water-d_598.html

Of the curve fits available, I found the best fits were to be with polynomial regression.

Quadratic Regression: y = a2 * x^2 + a1 * x + a0

y = -0.02736013 * x^2 + 4.059449 * x + 1407.3398

where y is the speed of sound in water (m/s) and x is the temperature (°C). The r^2 parameter is 0.9980768, which is pretty good given the raw data is only rounded the nearest integer. The Numworks also offers residual plot, which is the difference between predicted and given data. From the residual plot, the biggest difference was at x = 0.

Quartic Regression: y = a4 * x^4 + a3 * x^3 + a2 * x^2 + a1 * x + a0

If we are looking for better accuracy, we could use the quartic regression, where r^2 parameter is 0.99981. From the residual plot, the biggest difference was located at x = 60.

y = -7.955665 * 10^-7 * x^4 + 2.523644 * 10^-4 * x^3 – 0.05122094 * x^2 + 4.782147 * x + 1403.692

For a quick calculation, the quadratic equation could be sufficient enough.

Source

The Engineering ToolBox (2004). Water - Speed of Sound vs. Temperature. [online] Available at: https://www.engineeringtoolbox.com/sound-speed-water-d_598.html. Accessed May, 2025.

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

The author does not use AI engines and never will.