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Integral from -2 to 2 (integral from x^2 to 4 (integral from -sqrt (y-x^2) to sqrt (y-x^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...

Hey lrdheat,
in the Nspire you can divide the result by pi and then use approxFraction (with an approximation of 0.001).



Best,
Aries

\(\large \int _2^2 \int _{x^2}^4 \int _{-\sqrt{y-x^2}}^{+\sqrt{y-x^2}} \sqrt{x^2+z^2}\;dz\;dy\;dx\)

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*pi-1024)/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*pi-1024)/225

XCas> float([f1, f2]) // f2 gives correct result
→ [28.1720021403 , 8.06580915733]

With the bug, result have excess of (f1-f2)/2. Remove it, we have area:

XCas> simplify(176*pi/15 - (f1-f2)/2)
→ 128*pi/15

You can rewrite the initial integral using parity in z and get the right answer:
If you add parity in x, you will have to simplify the answer.
Multiple definite integrals are (too?) hard to integrate symbolically because the inner integrals have parameters and that prevent checking antiderivative discontinuities checks.

I think it is a real bug.

XCas> f := x^2*log(sqrt(y) + sqrt(y-x^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 1.4.9-57 (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(t1-t2, x=-2..2) * 1.     // 28.1720021403 bad
XCas> int(t1-t2, x= 0..2) * 2.     // 28.1720021403 bad
XCas> int(t1-t2, 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(Y-X*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

I don't think so:
If you plot(t1-t2,x=-2..2) the value of about 28 is credible.
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.

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.

Can it do (A+B)^3 factorials?
Like:

https://www.youtube.com/watch?v=dVs26SSUJSA

Thank you

(05-10-2020 09:06 PM)tom234 Wrote: [ -> ]Can it do (A+B)^3 factorials?
Like:

https://www.youtube.com/watch?v=dVs26SSUJSA

Thank you

Yep, sure

(05-11-2020 07:07 AM)Aries Wrote: [ -> ]

(05-10-2020 09:06 PM)tom234 Wrote: [ -> ]Can it do (A+B)^3 factorials?
Like:

https://www.youtube.com/watch?v=dVs26SSUJSA

Thank you

Yep, sure

So your saying HP Prime Solve in the math menu can solve xy^2?

(04-26-2020 06:53 PM)lrdheat Wrote: [ -> ]Integral from -2 to 2 (integral from x^2 to 4 (integral from -sqrt (y-x^2) to sqrt (y-x^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...