math question...numerical integration

[eLiu]

Hey folks...
So I'm supposed to numerically integrate this:

exp(-x*y*z), dxdydz. Bounds are 0 to 8 for all 3 vars.

What I've tried so far:
Integrate dx by hand...that's easy enough:
-(1-exp(-8*y*z))/y*z dydz.

Expand exp(-8*y*z), divide by yz, integrate again. You get an alternating series:
[512^n * (-1)^(n+1)]/(n^2 * n!). It converges to the desired accuracy in ~1420 runs...not exactly fast, and not exactly what my prof wants. But it yields the correct result.

Other things:
We want to get rid of the yz in the denominator, so take
x=8*e^u
y=8*e^v <--2 parameter change of variables

Result: 1-exp(-512*exp(u)*exp(v)) dudv; bounds are -infinity to 0 for both.

Now I'm stuck. I'm thinking I need to somehow reduce this to a single integral without the infinite bounds and I'll be golden...too bad I'm running dry on ideas 🙁 Ideally I'm aiming for a single var integration of a well behaved function so I can use the romberg method.

Edit: I thought of:
u= a - ln(b)
v= ln(b)

But then that yields an expression that I evaluate as infinity... booo and it had so much promise!

Thanks,
-Eric

 

[Mrvile]

omg my name is Eric too.

Sorry for spamming your thread.

 

[bonkers325]

sounds like triple integration. integrate wrt to z, then y, then x. you should end up with a numerical value

 

[eLiu]

Originally posted by: bonkers325
sounds like triple integration. integrate wrt to z, then y, then x. you should end up with a numerical value

lol...it can't be integrated analytically--not all the way through. There's a double integration algorithm in the textbook, but we aren't supposed to use it for this problem.

 

[chuckywang]

If you want a answer expressed as an alternating sum, then I got:

SUM(8^(3n)/(n^2*n!),n,1,infinity)

 

[bonkers325]

i guess you should use mathcad or matlab

 

[Gibson486]

uhhh...if you want to numercally intergarte this, why not just integrate it three times with respect to each variable?

edit:
oh...you mean you want the actual number?

 

[eLiu]

chuckywang, I already got that... 😉 I don't think my prof wants a series though, which is why I'm looking for an alternate method.

bonkers325 I've programmed an algorithm to use the trapezoid method w/romberg extrapolation in MATLAB. There's no built-in function to just get the answer. I've done the other hmwk problems with my function--like sqrt(sin(x)) from 0 to pi, the circumference & area of x^4 + y^4 = 1, sum from 1 to infinity of 1/n^(3/2)--it's just the zeta function, and some others.

Gibson486, it's not possible to integrate this by hand. It's one of those that has no solution, like integral sin(x)/x from 0 to pi. I can get one integral out of the way by hand (as indicated in OP), and I'm attempting to remove another integral through substitution right now...not working 🙁 I know it does work b/c my prof said it would, lol.

 

[chuckywang]

What exactly are you looking for? I thought you wanted the actual number expressed as a sum of the triple integral. Are you instead looking for a way to reduce this to a one integral problem? If so, you have to be able to take the definite integral of e^x/x from 0 to 8, which I don't think you can do.

 

[eLiu]

Originally posted by: chuckywang
What exactly are you looking for? I thought you wanted the actual number expressed as a sum of the triple integral. Are you instead looking for a way to reduce this to a one integral problem? If so, you have to be able to take the definite integral of e^x/x from 0 to 8, which I don't think you can do.

Indeed exp(x)/x from 0 to 8 is infinity. However note that what I have here is (1-exp(x))/x, which is integrable (by a computer at least). That's kinda one of the forms I expressed in the OP.

But yeah I'm looking to turn the triple integral into a single integral by analytic methods. That single integral will be evaluated w/a computer b/c it'll definitely be analytically impossible.

 

[chuckywang]

Originally posted by: eLiu

Originally posted by: chuckywang
What exactly are you looking for? I thought you wanted the actual number expressed as a sum of the triple integral. Are you instead looking for a way to reduce this to a one integral problem? If so, you have to be able to take the definite integral of e^x/x from 0 to 8, which I don't think you can do.

Indeed exp(x)/x from 0 to 8 is infinity. However note that what I have here is (1-exp(x))/x, which is integrable (by a computer at least). That's kinda one of the forms I expressed in the OP.

But yeah I'm looking to turn the triple integral into a single integral by analytic methods. That single integral will be evaluated w/a computer b/c it'll definitely be analytically impossible.

I don't think (1-e^x)/x has a definite integal either. You might be able to do some trick with your double integral like how you evaluate the integral of e^(x^2) from -infinity to infinity.