Integral CAS approximation
05-01-2021, 08:25 PM
Post: #1
 lrdheat Senior Member Posts: 866 Joined: Feb 2014
Integral CAS approximation
Curious concerning why the disagreement in approximation of an integral in CAS vs the same integral in HOME. When an exact integration is impossible for a function, should HOME be trusted to bring more accuracy? I was expecting an approximation in CAS would force a HOME type numerical approximation.

CAS approximation of integral from -1 to 1 of SQRT(1+x^3) produces 1.95275719206

In home, I get 1.95275723373 which I get on the WP 34S on a DM42 platform and 1.952757234 on a CASIO 9750GIII.
05-01-2021, 08:26 PM
Post: #2
 lrdheat Senior Member Posts: 866 Joined: Feb 2014
RE: Integral CAS approximation
This on the beta April 28...
05-02-2021, 05:51 AM
Post: #3
 robmio Member Posts: 172 Joined: Jan 2020
RE: Integral CAS approximation
You get identical results in both "HOME" and "CAS" if you use the extremes -1.0 and 1.0, instead of -1 and 1 (note the absence of the dot after -1 and 1). The same curious behavior can be found in "xCas" version 1.7
05-02-2021, 06:30 AM
Post: #4
 parisse Senior Member Posts: 1,327 Joined: Dec 2013
RE: Integral CAS approximation
HOME is using CAS for integral evaluation, but without trying an exact computation first (in other words it will call integrate(sqrt(1+x^3),x,-1.0,1.0)).
If you run integrate(sqrt(1+x^3),x,-1,1) in CAS, you get the initial integral expressed with other integrals because the CAS tried to get an exact answer, and this explains the small numeric differences, the general rule is that the precision is better without initial symbolic step (if the integral can not be expressed in closed form).
05-02-2021, 02:35 PM
Post: #5
 Albert Chan Senior Member Posts: 2,701 Joined: Jul 2018
RE: Integral CAS approximation
On HP emulator, integration seems work harder if we use pattern ∫(f(x), x= a .. b)

We integrate ∫(√(1+x^3), x= -1 .. 1) backwards (*), letting t = -x

Cas> expand(∫(√(1-t^3), t, -1, 1))

√2*2*1/5 + integrate(3/5/√(-t^3+1),t,-1,1)

Cas> float(Ans) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.95275719206

Cas> expand(∫(√(1-t^3), t = -1 .. 1))

√2*2*1/5 + 1/5*√π*Gamma(1/3)/Gamma(5/6) + integrate(3/5/√(-t^3+1),t,-1,0)

Cas> float(Ans) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.95275723373

(*) Cas knows ∫(√(1-t^3), t,0,1), but not its equivalent, ∫(√(1+x^3), x,-1,0)
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