Solving Integral Equations

Solving Integral Equations

Solving integral equations (for the variable A) of any of the following forms:

∫( f(x) dx, x = constant to x = A) = area

∫( f(x) dx, x = A to x = constant) = area

∫( f(x) dx, x = constant to x = constant) = A

is fairly easy with the solve feature on non-graphing scientific calculators such as:

Casio fx-115ES Plus 

Casio fx-991EX

Casio fx-991CW

TI-30X Pro Math Print

TI-36X Pro

The screen shots below show an example, from the Casio fx-991CW (Equation - Solver):

The function must have the variable x as the independent variable. We can designated any other variable to be limit to be solved for; I used the variable a.  

Here, we don't have to worry about the value of x when solving the equation because x is considered a dummy variable in the integral.  

Examples

Solve for A.

Example 1:

∫( e^(-x^2) dx, x = 0 to x = A) = 0.7

Casio fx-991EX:  

A = 0.8861430055,  L-R = 0

TI-30X Pro MathPrint:  

A = 0.8861430055201,  L-R = 0  (this solve took a little time)

Example 2:

∫( e^(-x^2) dx, x = A to x = 2) = 0.13

Casio fx-991EX:  

A = 1.014499933,  L-R = 0

TI-30X Pro MathPrint:  

A = 1.014499933366,  L-R = 0  

Example 3:

∫( A*x + 1 dx, x = 1 to x = 5 ) = 10

Casio fx-991EX:  

A = 0.4,  L-R = 0

TI-30X Pro MathPrint:  

A = 0.4,  L-R = 0  

Eddie 

All original content copyright, © 2011-2023.  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-991EX and fx-115ES Plus: Velocity Conversions

Casio fx-991EX and fx-115ES Plus: Velocity Conversions

The Casio fx-991EX Classwiz and Casio fx-115ES Plus calculators, among others, have 40 conversions in various areas. 

Keystrokes

Keystrokes: Casio fx-115ES Plus

[ SHIFT ] [ 8 ], enter the required two digit code

The conversions we'll be working with this blog entry:

ft → m, 03
m → ft, 04
yd → m, 05
m → yd, 06
mile → km, 07
km → mile, 08
km/hr → m/s, 19
m/s → km/hr, 20

Keystrokes:  Casio fx-991EX Classwiz

[ SHIFT ] [ 8 ].    The conversions will be grouped by category (length, area, volume, etc).

ft → m, 1, 3
m → ft, 1, 4
yd → m, 1,5
m → yd, 1,6
mile → km, 1, 7
km → mile, 1, 8
km/hr → m/s, ↓, 1, 1
m/s → km/hr, ↓, 1, 2

km/hr  (kilometers per hour) vs mph (miles per hour)

The calculators mentioned does not have a direct conversion between km/hr and mph.  However, because the time frame is the same (per hour), we can use the km/mile conversions (since 1 hour = 1 hour and no time conversion part is required, only the length).

Example 1:  150 km/hr to mph

150 km>mile returns 93.20567884

150 km/hr = 9320567884 mph

Example 2:  65 mph to km/hr

65 mile>km = 104.60736

65 mph = 104.60736 km/hr

More Examples

Example 3:  5.8 ft/s to m/s

Again, since the time portion is the same (second to second), we only have to worry about length.

5.8 ft>m returns 1.76784

5.8 ft/s = 1.76784 m/s

Example 4:  110 ft/s to km/hr

Now we are converting both length and time.   There are no direct time conversions on the calculators featured, so we have to use the manual conversion. 

Remember that 1 hr = 3600 s and 1 km = 1000 m. To convert from m to km, divide by 1000 and to convert from 1/s to 1/hr, multiply by 3600 (note: 1/s to 1/hr)

(110 ft>m) ÷ 1000 × 3600 returns 120.7008

110 ft/s = 120.7008 km/hr

Example 5:  40 m/s to km/hr

Here is another example where we both have to convert the length (m to km) and time component (1/s to 1/hr).   This is similar to the Example 3.  However, the Casio calculators featured on this blog has conversions between m/s and km/hr (velocity section).

40 m/s>kh/hr returns 144 km/hr

40 m/s = 144 km/hr

Example 6:  100 ft^2 to m^2

The calculators have two area conversions (acre/m^2).   However, we can use the same length conversion twice to make other areas.

100 ft>m ft>m returns 9.290304

100 ft^2 = 9.290304 m^2

Example 7:  1 mi to yd by using the conversions available

(1 mi>km × 1000) m>yd returned 1760

(1 km = 1000 m)

1 mi = 1760 yd

A Cautionary Tale

Example 8:  1 acre to ft^2

1 acre>m^2 m>ft m>ft returns 43559.99545

However, as defined, 1 acre = 43560 ft^2. 

Lesson:  use caution when chaining conversions, 100% accuracy isn't guaranteed.

Eddie

All original content copyright, © 2011-2020.  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 Calculators and the Percent Key

Casio Calculators and the Percent Key

Introduction

The percent function on Casio calculators varies based on the calculator itself.  Let's compare six key algorithms on six Casio calculators, five current and one from the past.  We will go from the most consistent behavior to the least.

The calculators used in this study are:

SL-300VC  (four function calculator)
fx-260 Solar II  (scientific calculator - one line display)
fx-300MS 2nd Edition (scientific calculator - one line display)
fx-115ES Plus  (scientific calculator - textbook display)
fx-991EX Classwiz (scientific calculator - textbook display)
fx-115D  (scientific calculator introduced in 1991 - one line display)

Let A and B be real numbers.  In the following examples, required press of the [SHIFT] key are implied as required. 

Algorithm 1:  Multiply

Keystrokes:  A [ × ] B [ % ]

Let A = 82, B = 30

SL-300VC:  24.6 
fx-260 Solar II:  24.6
fx-300MS 2nd Edition:  24.6
fx-115ES Plus**:  123/5 ([ S<>D] 24.6) 
fx-991EX Classwiz**:  123/5  ( [ S<>D] 24.6)
fx-115D:  24.6

A [ × ] B  [ % ] calculates A * B/100.  No surprise that the results will be consistent across the board. 

**Pressing equals ( [ = ] ) is required to complete calculations involving percent calculations for the calculators with textbook display. 

Algorithm 2:  Division

Keystrokes:  A [ ÷ ] B [ % ]

Let A = 82, B = 30

SL-300VC:  273.33333 
fx-260 Solar II:  273.3333333
fx-300MS 2nd Edition:  273.3333333
fx-115ES Plus:  820/3 ([S <> D] 273.3333333) 
fx-991EX Classwiz:  820/3 ([S <> D] 273.3333333)
fx-115D: 273.3333333

A [ ÷ ] B calculates A  * 100/B.   Results are consistent across the board.

Algorithm 3:  Adding Percents

Keystrokes:  A [ × ] B [ % ] [ + ]  

A = 57, B = 11

SL-300VC: 63.27 
fx-260 Solar II: 63.27
fx-300MS 2nd Edition:  63.27
fx-115D:  63.27 

This calculates A * (1 + B/100)

If you tried the exact algorithm with the textbook style calculators like the fx-115ES Plus and the fx-991EX Classwiz, you get an error.  Use the following instead:   A [ (  ] 1 [ +  ] B [ % ] [  ) ] [ = ]

Algorithm 4:  Subtracting Percents

Keystrokes:  A [ - ] B [ % ] [ + ] 

A = 57, B = 11

SL-300VC: 50.73
fx-260 Solar II: 50.73
fx-300MS 2nd Edition:  50.73
fx-115D:  50.73 

This calculates A * (1 - B/100)

If you tried the exact algorithm with the textbook style calculators like the fx-115ES Plus and the fx-991EX Classwiz, you get an error.  Use the following keystroke algorithm instead:   A [ (  ] 1 [ -  ] B [ % ] [  ) ] [ = ]

Algorithm 5:  A - B%

Keystrokes:  A [ - ] B [ % ]

SL-300VC: 12.5 
fx-260 Solar II: 12.5
fx-300MS 2nd Edition: 12.5
fx-115ES Plus:  1784/25 ( [S<>D] 71.36) 
fx-991EX Classwiz:  1784/25 ( [S<>D] 71.36) 
fx-115D: 12.5

We see some difference on how this algorithm is calculated and it depends on the type of calculator used.

For four-function basic calculators and one-line scientific calculators, A - B% calculates the percent change:  (A - B)/B * 100

For the  textbook display calculators, the B is merely divided by 100 and subtracted from A:  A - B/100

Algorithm 6:  A + B%

Keystrokes:  A [ + ] B [ % ]

SL-300VC: 200
fx-260 Solar II: 300
fx-300MS 2nd Edition: 300
fx-115ES Plus:  201/2 ( [S<>D] 100.5) 
fx-991EX Classwiz:  201/2 ( [S<>D] 100.5) 
fx-115D: 300

This is the most inconsistent.  Quite honestly, I don't recommend this algorithm. 

What I learned:

1.  Check your manual on how to use the percent function ( % ), not matter what calculator you use. 

2.  Calculators with textbook display, the percent function ( % ) merely divides the argument by 100.

3.  This is an sample of six calculators, and as you can see, your mileage may vary. 

Have fun calculating,

Eddie

All original content copyright, © 2011-2019.  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.

Solar Scientific Calculators: Dealing with Integrals with Infinite Limits

Solar Scientific Calculators:  Dealing with Integrals with Infinite Limits

Integrals with Infinite Limits

Today's post deals with integrals with infinite limits in the forms:

∫( f(x) dx, x = a to x = ∞)

∫( f(x) dx, x = -∞ to x = ∞)

∫( f(x) dx, x = -∞ to x = a)

One method to deal with these integrals, as suggested by W.A.C. Mier-Jedrzejowicz Ph. D. (see the source), is to use the substitution

x = tan θ

Then:

dx = dθ/cos^2 θ

and θ = atan x.

Also, as x approaches π/2, tan x approaches +∞.

And, as x approaches -π/2, tan x approaches -∞.

With the substations, let's test four integrals on four solar-powered scientific calculators:

1.  Casio fx-991EX Classwiz
2.  Sharp EL-W516T
3.  Texas Instruments TI-36X Pro
4.  Casio fx-115ES Plus

Set the calculator to radians mode. 

Example 1:  ∫(1/x^2 dx, x = 1 to x = ∞) = 1

∫(1/x^2 dx, x = 1 to x = ∞)

with the substitutions x = tan θ and dx = dθ/(cos^2 θ):

∫( 1/tan^2 θ * dθ/cos^2 θ, θ = atan 1 to θ = π/2)

∫( 1/sin^2 θ * dθ, θ = atan 1 to θ = π/2)

We can evaulate the integral straight away.  Here are the results:

1.  Casio fx-991EX Classwiz
Time: 1.37 seconds
Answer: 1

2.  Sharp EL-W516T
Time: 38 seconds
Answer: 1

3.  Texas Instruments TI-36X Pro
Time: 4.5 seconds
Answer: 1

4.  Casio fx-115ES Plus
Time: 4.2 seconds
Answer: 1

A promising start.

Example 2:  ∫(e^(-0.5*x^2), x = 0 to x = ∞) ≈ 1.25331413732

∫(e^(-0.5*x^2), x = 0 to x = ∞)

with the substituions, this becomes:

∫(e^(-0.5 * tan^2 θ)/cos^2 θ dθ, θ = atan 0 to θ = π/2)

atan 0 = 0

But look at the denominator, we have cos^2 θ.  Since cos^2 π/2 = 0, there will be a problem.  Let's use an approximation of π/2 of 1.5708.

∫(e^(-0.5 * tan^2 θ)/cos^2 θ dθ, θ = 0 to θ = 1.5708)

Here are the results:

1.  Casio fx-991EX Classwiz
Time: 15.4 seconds
Answer: 1.253314137

2.  Sharp EL-W516T
Time: 1 minute, 8 seconds
Answer: errors out

3.  Texas Instruments TI-36X Pro
Time: 36 seconds
Answer: 1.253314138

4.  Casio fx-115ES Plus
Time: 1 minute, 6.8 seconds
Answer: 1.253314137

Example 3:  ∫(x^2*e^-x dx, x = 0 to x = ∞) = 2

∫(x^2*e^-x dx, x = 0 to x = ∞)

with the substitutions and simplification, we get:

∫( (sin^2 θ * e^(-tan θ))/cos^4 θ dθ, θ = 0 to θ = π/2)

Like the last situation, there is a potential problem with the denominator.  Let's see if we can use an approximation of π/2, this time using 1.57 in hopes to cut the calculation time down.

∫( (sin^2 θ * e^(-tan θ))/cos^4 θ dθ, θ = 0 to θ = 1.57)

Here are the results:

1.  Casio fx-991EX Classwiz
Time: 27 seconds
Answer: 2

2.  Sharp EL-W516T
Time: 1 minute, 34 seconds
Answer: 1.999999999

3.  Texas Instruments TI-36X Pro
Time: 1 minute, 9 seconds
Answer: 2

4.  Casio fx-115ES Plus
Time: 1 minute, 6.8 seconds
Answer: 1.253314137

Example 4:  ∫( e^-x/x^2 dx, x = 1 to x = ∞) ≈ 0.148495506776

∫( e^-x/x^2 dx, x = 1 to x = ∞)

with the substitutions and simplification, we get:

∫( e^(-tan θ)/sin^2 θ dθ, θ = 0 to θ = π/2)

I'm going to use the 1.57 approximation again and set the integral as:

∫( e^(-tan θ)/sin^2 θ dθ, θ = 0 to θ = 1.57)

Here are the results:

1.  Casio fx-991EX Classwiz
Time: errors out immediately
Answer: N/A

2.  Sharp EL-W516T
Time: 1 minute, 6 seconds
Answer: error

3.  Texas Instruments TI-36X Pro
Time: 7 seconds
Answer: 0.148495519

4.  Casio fx-115ES Plus
Time: errors out after 1 second
Answer: N/A

Some Observations

1.  Not all calculations of improper integrals will be successful.

2.  Out of the four calculators tested, from the four calculations:  the Casio fx-991ES is the fastest, but I found the most successful with the Texas Instruments TI-36X Pro.

3.  Be ready to spend a little for calculations by using this method.

Source:

Mier-Jedrzejowic, W.A.C. Ph.D.  Extend Your 41  London, UK 1985  ISBN 0-9510733-0-03

Eddie

All original content copyright, © 2011-2019.  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.  Please contact the author if you have questions.

Favorite Solar Calculators – Six Years Later

Favorite Solar Calculators – Six Years Later

Favorite Solar Calculators – Reading that Blog Entry Today

On June 10, 2012, I listed some of my favorite solar calculators of all time: 

https://edspi31415.blogspot.com/2012/06/my-favorite-solar-calculators-of-all.html

I still love the TI-25X Solar, I wish the screens weren’t so damn fragile!  I went through two of them. 

I have two fx-115 ES Plus calculators, one gray and one black.  The black looks much better!

Maybe one day I’ll get the BA 35 Solar again from eBay.  Maybe I will give the Casio FC-200V a second shot. 

As of March 2018, Casio fx-3650pII is the current edition of the fx-3650P.  The fx-3650PII is in the shape of the fx-115ES/991ES/300ES/82ES Plus.  That’s about it from what I can get from research:  still four program areas with 360 steps and 7 variables.

Info on the current fx-3650PII:  http://www.casio-intl.com/asia/en/calc/products/fx-3650PII/

No, Casio still currently sells the fx-3650PII only outside the United States, which means for us Americans, we have to order online.  An open invite to Casio to stock the office stores and university stores with this model in the States is extended. 

The current Sharp EL-W516 edition, the EL-516WT, eliminated the formula storage, eliminated the catalog, and reduced the definable keys from 4 to 3, I am not happy about that.

My review on the EL-W516T:  https://edspi31415.blogspot.com/2017/07/review-sharp-el-w516t.html

Three More Inductees

I have three more to add to the list I posted in 2012:

Casio fx-991EX Classwiz – 2015 – Present

The Classwiz model is the next step in Casio solar powered calculators.  They have an icon menu, like their graphing calculators, and there is an [OPTN] key that allows users to specific mode-specific functions.  Modes includes computation (COMP), complex numbers (arithmetic, polar/rectangular conversions), base integers, matrices up to 4 x 4, vectors, statistics, distributions, basic spreadsheets up to 5 x 45, equation solving, inequalities and ratios.  The newest feature is the QR function that will take a screen shot of the calculator, or in some cases, statistical graphs, which can be retrieved from the Casio QR Website or app.

See my review of the Classwiz here:  https://edspi31415.blogspot.com/2015/11/casio-fx-991ex-classwiz-review.html

I can only hope the next iteration the Classwiz and the fx-3650P or fx-50FII merge so programming will be included.

TI-30 SLR+ – 1987 – 1990s

This is a bigger version of the TI-25X Solar (TI-30X Solar internationality).  This calculator completely runs on solar and light power.  What I look about this better than the TI-25X are the keys and the fact the display isn’t so fragile.  See my detailed blog of it here:  https://edspi31415.blogspot.com/2014/09/ti-30-slr-and-memories-of-school.html

Casio fx-260 Solar II (we can included the Casio fx-260 Solar, fx-82 Solar, fx- 82 Solar II)

Original fx-260 Solar/fx-82 Solar:  2000s – present

Current fx-260 Solar II/fx-82 Solar II: 2017 – present

I don’t know what I was thinking when I skipped this model last time.  The fx-260 offers a wide variety of functions:  trigonometry, logarithms, fractions, degrees and degrees-minute-second conversions, polar/rectangular conversions, and one variable statistics.  This is great calculator for those who want a small, compact, calculator that attacks the basics.  The fx-260 series is completely solar powered.  Pictured is the newer fx-260 Solar II, which the mode markers have been mode to the back of the calculator case.  Can’t beat the classics. 

Review:  https://edspi31415.blogspot.com/2017/03/review-casio-fx-260-solar-ii-fx-82.html

Eddie

This blog is property of Edward Shore, 2018.