Dividing triple integral to evaluate the integrals in closed-form
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I have the following triple integral
$$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right),dx_2dy_1dy_2$$
I want to divide the integrals into intervals to evaluate it. What I did was that I divided the inner most integral into two interval as following
$$int_x_2=0^fracz,y_1y_2y_1+y_2expleft(-x_2right),dx_2 + int_x_2=fracz,y_1y_2y_1+y_2^zy_2expleft(-y_1(z-fracx_2y_2)right),dx_2$$
Although this allows us to evaluate the integral over $x_2$ easily, the integration over $y_1$ and $y_2$ will be harder.
My actual problem is a little more complicated, and this question is used to capture the essence of the problem. Thus, I am not looking for a solution for the integrals as they are, but rather I need a method where I can divide all the integrals (not just the inner most one) into intervals that makes the evaluation of all integrals much simpler. Is this feasible?
EDIT: My actual problem is the following
$$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right)e^-y_1,e^-y_2,dx_2dy_1dy_2$$
calculus integration statistics probability-distributions
asked Aug 8 at 20:10
BlackMath
948
add a comment |Â
up vote
1
down vote
favorite
I have the following triple integral
$$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right),dx_2dy_1dy_2$$
I want to divide the integrals into intervals to evaluate it. What I did was that I divided the inner most integral into two interval as following
$$int_x_2=0^fracz,y_1y_2y_1+y_2expleft(-x_2right),dx_2 + int_x_2=fracz,y_1y_2y_1+y_2^zy_2expleft(-y_1(z-fracx_2y_2)right),dx_2$$
Although this allows us to evaluate the integral over $x_2$ easily, the integration over $y_1$ and $y_2$ will be harder.
My actual problem is a little more complicated, and this question is used to capture the essence of the problem. Thus, I am not looking for a solution for the integrals as they are, but rather I need a method where I can divide all the integrals (not just the inner most one) into intervals that makes the evaluation of all integrals much simpler. Is this feasible?
EDIT: My actual problem is the following
$$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right)e^-y_1,e^-y_2,dx_2dy_1dy_2$$
calculus integration statistics probability-distributions
asked Aug 8 at 20:10
BlackMath
948
Are you familiar with the XY problem?
– Robert Howard
Aug 8 at 20:20
I am not familiar with it. I just read about it though, and I am not sure what you are implying. If you are talking about why I didn't post my original problem, it is because I think it won't change anything in the approach. But here it is in case you are wondering and it is important to the solution $$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right)e^-y_1,e^-y_2,dx_2dy_1dy_2$$ I hope this provides enough information.
– BlackMath
Aug 8 at 20:34
I didn't mean to imply anything; my apologies. It's not uncommon for people on this site to have burrowed their way into a problem and ask for help on a tiny piece of it that ultimately doesn't have much to do with their end goal. When you mentioned "[your] actual problem," I wondered if that was what had happened here, but I see now that that's not the case. Within reason, more detail and context never hurt!
– Robert Howard
Aug 8 at 21:36
I admit I don't know how to solve your problem, but hopefully what you've provided is enough that someone more knowledgeable than me can help you with it!
– Robert Howard
Aug 8 at 21:37
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have the following triple integral
$$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right),dx_2dy_1dy_2$$
I want to divide the integrals into intervals to evaluate it. What I did was that I divided the inner most integral into two interval as following
$$int_x_2=0^fracz,y_1y_2y_1+y_2expleft(-x_2right),dx_2 + int_x_2=fracz,y_1y_2y_1+y_2^zy_2expleft(-y_1(z-fracx_2y_2)right),dx_2$$
Although this allows us to evaluate the integral over $x_2$ easily, the integration over $y_1$ and $y_2$ will be harder.
My actual problem is a little more complicated, and this question is used to capture the essence of the problem. Thus, I am not looking for a solution for the integrals as they are, but rather I need a method where I can divide all the integrals (not just the inner most one) into intervals that makes the evaluation of all integrals much simpler. Is this feasible?
EDIT: My actual problem is the following
$$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right)e^-y_1,e^-y_2,dx_2dy_1dy_2$$
calculus integration statistics probability-distributions
asked Aug 8 at 20:10
BlackMath
948
I have the following triple integral
$$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right),dx_2dy_1dy_2$$
I want to divide the integrals into intervals to evaluate it. What I did was that I divided the inner most integral into two interval as following
$$int_x_2=0^fracz,y_1y_2y_1+y_2expleft(-x_2right),dx_2 + int_x_2=fracz,y_1y_2y_1+y_2^zy_2expleft(-y_1(z-fracx_2y_2)right),dx_2$$
Although this allows us to evaluate the integral over $x_2$ easily, the integration over $y_1$ and $y_2$ will be harder.
My actual problem is a little more complicated, and this question is used to capture the essence of the problem. Thus, I am not looking for a solution for the integrals as they are, but rather I need a method where I can divide all the integrals (not just the inner most one) into intervals that makes the evaluation of all integrals much simpler. Is this feasible?
EDIT: My actual problem is the following
$$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right)e^-y_1,e^-y_2,dx_2dy_1dy_2$$
calculus integration statistics probability-distributions
asked Aug 8 at 20:10
BlackMath
948
edited Aug 8 at 20:43
asked Aug 8 at 20:10
BlackMath
948
asked Aug 8 at 20:10
BlackMath
948
asked Aug 8 at 20:10
BlackMath
948
948
Are you familiar with the XY problem?
– Robert Howard
Aug 8 at 20:20
I am not familiar with it. I just read about it though, and I am not sure what you are implying. If you are talking about why I didn't post my original problem, it is because I think it won't change anything in the approach. But here it is in case you are wondering and it is important to the solution $$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right)e^-y_1,e^-y_2,dx_2dy_1dy_2$$ I hope this provides enough information.
– BlackMath
Aug 8 at 20:34
I didn't mean to imply anything; my apologies. It's not uncommon for people on this site to have burrowed their way into a problem and ask for help on a tiny piece of it that ultimately doesn't have much to do with their end goal. When you mentioned "[your] actual problem," I wondered if that was what had happened here, but I see now that that's not the case. Within reason, more detail and context never hurt!
– Robert Howard
Aug 8 at 21:36
I admit I don't know how to solve your problem, but hopefully what you've provided is enough that someone more knowledgeable than me can help you with it!
– Robert Howard
Aug 8 at 21:37
add a comment |Â
Are you familiar with the XY problem?
– Robert Howard
Aug 8 at 20:20
I am not familiar with it. I just read about it though, and I am not sure what you are implying. If you are talking about why I didn't post my original problem, it is because I think it won't change anything in the approach. But here it is in case you are wondering and it is important to the solution $$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right)e^-y_1,e^-y_2,dx_2dy_1dy_2$$ I hope this provides enough information.
– BlackMath
Aug 8 at 20:34
I didn't mean to imply anything; my apologies. It's not uncommon for people on this site to have burrowed their way into a problem and ask for help on a tiny piece of it that ultimately doesn't have much to do with their end goal. When you mentioned "[your] actual problem," I wondered if that was what had happened here, but I see now that that's not the case. Within reason, more detail and context never hurt!
– Robert Howard
Aug 8 at 21:36
I admit I don't know how to solve your problem, but hopefully what you've provided is enough that someone more knowledgeable than me can help you with it!
– Robert Howard
Aug 8 at 21:37
Are you familiar with the XY problem?
– Robert Howard
Aug 8 at 20:20
Are you familiar with the XY problem?
– Robert Howard
Aug 8 at 20:20
I am not familiar with it. I just read about it though, and I am not sure what you are implying. If you are talking about why I didn't post my original problem, it is because I think it won't change anything in the approach. But here it is in case you are wondering and it is important to the solution $$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right)e^-y_1,e^-y_2,dx_2dy_1dy_2$$ I hope this provides enough information.
– BlackMath
Aug 8 at 20:34
I am not familiar with it. I just read about it though, and I am not sure what you are implying. If you are talking about why I didn't post my original problem, it is because I think it won't change anything in the approach. But here it is in case you are wondering and it is important to the solution $$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right)e^-y_1,e^-y_2,dx_2dy_1dy_2$$ I hope this provides enough information.
– BlackMath
Aug 8 at 20:34
I didn't mean to imply anything; my apologies. It's not uncommon for people on this site to have burrowed their way into a problem and ask for help on a tiny piece of it that ultimately doesn't have much to do with their end goal. When you mentioned "[your] actual problem," I wondered if that was what had happened here, but I see now that that's not the case. Within reason, more detail and context never hurt!
– Robert Howard
Aug 8 at 21:36
I didn't mean to imply anything; my apologies. It's not uncommon for people on this site to have burrowed their way into a problem and ask for help on a tiny piece of it that ultimately doesn't have much to do with their end goal. When you mentioned "[your] actual problem," I wondered if that was what had happened here, but I see now that that's not the case. Within reason, more detail and context never hurt!
– Robert Howard
Aug 8 at 21:36
I admit I don't know how to solve your problem, but hopefully what you've provided is enough that someone more knowledgeable than me can help you with it!
– Robert Howard
Aug 8 at 21:37
I admit I don't know how to solve your problem, but hopefully what you've provided is enough that someone more knowledgeable than me can help you with it!
– Robert Howard
Aug 8 at 21:37
add a comment |Â
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Are you familiar with the XY problem?
– Robert Howard
Aug 8 at 20:20
I am not familiar with it. I just read about it though, and I am not sure what you are implying. If you are talking about why I didn't post my original problem, it is because I think it won't change anything in the approach. But here it is in case you are wondering and it is important to the solution $$int_y_1=0^inftyint_y_2=0^inftyint_x_2=0^zy_2expleft(-minleft[x_2,,y_1(z-fracx_2y_2)right]right)e^-y_1,e^-y_2,dx_2dy_1dy_2$$ I hope this provides enough information.
– BlackMath
Aug 8 at 20:34
I didn't mean to imply anything; my apologies. It's not uncommon for people on this site to have burrowed their way into a problem and ask for help on a tiny piece of it that ultimately doesn't have much to do with their end goal. When you mentioned "[your] actual problem," I wondered if that was what had happened here, but I see now that that's not the case. Within reason, more detail and context never hurt!
– Robert Howard
Aug 8 at 21:36
I admit I don't know how to solve your problem, but hopefully what you've provided is enough that someone more knowledgeable than me can help you with it!
– Robert Howard
Aug 8 at 21:37