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.
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.