can I trust this numerical triple integral from Matlab?
Computational Science people:
I originally posted this question at Math Stack Exchange and someone commented that I might get "much better" answers here:
I am a novice at numerical methods and Matlab. I am attempting to evaluate the following sum of two triple integrals (it can obviously be written more simply, but you still cannot evaluate it symbolically (?)). I am having trouble getting the $\LaTeX$ to work here, so I reluctantly broke it up into pieces here: I want to find the sum of
$$\frac{2}{((1/0.3) - 1)^2}\left(\int_1^{1/0.3}\int_1^{r_1}\int_0^{r_1-r_0}F_1(r_0,r_1,t)\exp(-\frac{(0.3)^2 t^2}{4})\,dt\,dr_0\,dr_1 \right),$$
and
$$\frac{2}{((1/0.3) - 1)^2}\left(\int_1^{1/0.3}\int_1^{r_1}\int_{r_1-r_0}^{r_1+r_0} F_2(r_0,r_1,t)\exp(-\frac{(0.3)^2 t^2}{4})\,dt\,dr_0\,dr_1 \right),$$
where
$$F_1(r_0,r_1,t)=\frac{t^2 r_0^3*(0.3)^3}{2r_1^3\sqrt{\pi}}$$
and
$$F_2(r_0,r_1,t)=\frac{(0.3)^3\pi^{3/2}(r_0+r_1-t)^4 (t^2+2t(r_0+r_1)-3(r_1-r_0)^2)^2}{288(\frac{4}{3}\pi r_0^3)(\frac{4}{3}\pi r_1^3)}.$$
EDIT (March 2 2013): Someone responded that they got Mathematica to do the integrals symbolically. I just attempted to do this (with simplified versions of the integrals) and Mathematica could only do the outer two of the first one, and stalled on the second one. 402
I would appreciate some help. Here is what I did.:
I attempted to evaluate
$$\int_1^2 \int_1^{r_2} \int_0^{r_2-r_1} \frac{r_1^3 t^2 \exp(-t^2)}{r_2^3}\,dt\,dr_1\,dr_2$$ via
Integrate[r1^3/r2^3t^2Exp (-t^2), {t, 0, r2 - r1}, {r1, 1, r2}, {r2, 1, 2}]
and Mathematica returns (I had trouble with the $\LaTeX$ here because the result is long. I broke it into two equations. if anyone knows a good way to display this please tell me):
$$\int_1^2 \frac{1}{64r2^2} e^{-1-r2^2}(2e^{2r2}(25+r2(19+2r2(1+r2)))-$$
$$e^{1+r2^2}(32r2(2+r2^2)) +\sqrt{\pi}(11+4r2^2(9+r2^2))\operatorname{Erf}[1-r2])\,dr2. 402