DM 41X: Doble integral using different modules fails

[Albert Chan]

(09-29-2024 08:08 PM)rawi Wrote:  I tested the evaluation of the double integral of the following function:
e^(-x^2/(y^2+y*x+1)) with limits for x and y from 0 to 1.
HP Prime delivers the result very fast: 0.822832

I was curious if there is a way to avoid double integral, and still get a good estimate.

For x=0..1, y=0..1 --> Denominator k = y^2+y*x+1 = 1..3

Assume k is just a constant. This reduced ∫∫ to ∫

∫(e^(-x^2/k), x=0..1) = √k * ∫e^(-t^2), t=0 .. 1/√k) = √(k*pi)/2 * erf(1/√k) = √(k*pi) * cdf(0 .. √(2/k))

I = ∫(e^(-x²/k), x=0..1) = √(2*pi) * cdf(0 .. z)/z      , where z = √(2/k)

Mean denominator
= ∫∫(y²+y*x+1), x=0..1, y=0..1)
= (∫(y²+1), y=0..1) + (∫x, x=0..1)*(∫y, y=0..1)
= (1/3 + 1) + 1/2*1/2
= 19/12

z = √(2/k) = √(24/19) = 1.1239

Since this is only rough estimate, I don't bother with interpolation.
We just look up from statistic book, cdf(0 .. z) where z = 1.12 and 1.13

I(z=1.12) = 2.5066 * 0.3686 / 1.12 = 0.8249
I(z=1.13) = 2.5066 * 0.3708 / 1.13 = 0.8225