Triple Integral

04262020, 06:53 PM
Post: #1




Triple Integral
Integral from 2 to 2 (integral from x^2 to 4 (integral from sqrt (yx^2) to sqrt (yx^2) of sqrt (x^2 + z^2))) dz dy dx
Stewart Calculus Book Alternative Edition 7E (soft cover) page 1020 gives this example with an answer of 128*pi/15 The Prime G2 gives a number of warnings, produced 176*pi/15 in CAS In home, it gave an error of infinite result. My TI Nspire CX II produces the decimal equivalent of 128*pi/15 (~26.8083) My TI Nspire CAS emulator does not come up with the exact result, but does produce the correct decimal equivalent. Prime has difficulty with this example... 

04282020, 01:43 PM
Post: #2




RE: Triple Integral  
04282020, 08:27 PM
Post: #3




RE: Triple Integral
\(\large \int _2^2 \int _{x^2}^4 \int _{\sqrt{yx^2}}^{+\sqrt{yx^2}} \sqrt{x^2+z^2}\;dz\;dy\;dx\)
I isolated the problem ... XCas> w := sqrt(y  x^2) XCas> expand(int(sqrt(x^2+z^2), z = w .. w)) x^2*ln(sqrt(y)+sqrt(x^2+y))/2  x^2*ln(sqrt(y)(sqrt(x^2+y)))/2 + sqrt(y)*sqrt(x^2+y) XCas were evaluating the first term wrong. Converting to 2nd term, we have: XCas> expand([sqrt(y)+w , 1] .* (sqrt(y)w)) → [x^2 , sqrt(y)(sqrt(x^2+y))] These 2 integrals should gives the same answer (Wolfram Alpha returns 8.06581 for both) XCas> f1 := simplify(int(int(x^2*log(sqrt(y)+w), y = x^2 .. 4), x = 2 .. 2)) → (1920*ln(2)+1920*pi1024)/225 XCas> f2 := simplify(int(int(x^2*ln(x^2)  x^2*ln(sqrt(y)w), y=x^2 .. 4), x=2 .. 2)) → (1920*ln(2)+480*pi1024)/225 XCas> float([f1, f2]) // f2 gives correct result → [28.1720021403 , 8.06580915733] With the bug, result have excess of (f1f2)/2. Remove it, we have area: XCas> simplify(176*pi/15  (f1f2)/2) → 128*pi/15 

04292020, 06:15 AM
Post: #4




RE: Triple Integral
You can rewrite the initial integral using parity in z and get the right answer:
Code: 2*int(int(int(sqrt(x^2+z^2), z = 0 .. sqrt(y  x^2)),y,x^2,4),x,2,2) Code: 4*int(int(int(sqrt(x^2+z^2), z = 0 .. sqrt(y  x^2)),y,x^2,4),x,0,2) 

04292020, 12:47 PM
(This post was last modified: 04292020 01:43 PM by Albert Chan.)
Post: #5




RE: Triple Integral
I think it is a real bug.
XCas> f := x^2*log(sqrt(y) + sqrt(yx^2)) XCas> g := int(f, y = x^2 .. 4) // keep running this single integral, we get XCas> subst(g, x=1.5) // either 3.76506896552 or 13.7445470629 XCas 1.4.957 (Win32) gives 2 different answer from the same integral, g = t1 ± t2 XCas> t1 := 4*x^2*ln(sqrt(x^2+4)+2)  x^4*ln(abs(x)) XCas> t2 := x^2*sqrt(x^2+4)  x^4*ln(x^2)/4 + x^4*ln(abs(x^2+4*sqrt(x^2+4)8))/4 XCas> int(t1+t2, x=2..2) * 1. // 8.06580915733 ok XCas> int(t1+t2, x= 0..2) * 2. // 8.06580915733 ok XCas> int(t1+t2, x=2..0) * 2. // 8.06580915733 ok XCas> int(t1t2, x=2..2) * 1. // 28.1720021403 bad XCas> int(t1t2, x= 0..2) * 2. // 28.1720021403 bad XCas> int(t1t2, x=2..0) * 2. // 28.1720021403 bad Edit: numerical confirmation from EMU71 >10 P=.000001 >20 DEF FNF(X,Y)=X^2*LN(SQRT(Y)+SQRT(YX*X)) >30 DEF FNG(X)=INTEGRAL(X*X,4,P,FNF(X,IVAR)) >40 DEF FNH(A,B)=INTEGRAL(A,B,P,FNG(IVAR)) >RUN >FIX 6 >FNH(2,2), FNH(2,0)*2, FNH(0,2)*2 8.065809 8.065809 8.065809 

04292020, 01:57 PM
Post: #6




RE: Triple Integral
I don't think so:
Code:
BTW, 1.4.9 is relatively old, if you want to report bugs in Xcas, please check with the latest version :) I have improved embedded assumptions checking for some simplifications, now the initial triple integral returns the correct exact value. 

04292020, 02:59 PM
Post: #7




RE: Triple Integral
Thanks!
I was a little more surprised at this problem’s failure in “home”. Does your adjustment produce a good result there? Thanks for your fantastic work and interest in these areas. 

05102020, 09:06 PM
Post: #8




RE: Triple Integral  
05112020, 07:07 AM
Post: #9




RE: Triple Integral
(05102020 09:06 PM)tom234 Wrote: Can it do (A+B)^3 factorials? Yep, sure 

05112020, 10:35 AM
Post: #10




RE: Triple Integral
(05112020 07:07 AM)Aries Wrote:(05102020 09:06 PM)tom234 Wrote: Can it do (A+B)^3 factorials? So your saying HP Prime Solve in the math menu can solve xy^2? 

05112020, 02:18 PM
Post: #11




RE: Triple Integral
(04262020 06:53 PM)lrdheat Wrote: Integral from 2 to 2 (integral from x^2 to 4 (integral from sqrt (yx^2) to sqrt (yx^2) of sqrt (x^2 + z^2))) dz dy dx 

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