Dubbelintegral

[AnyHelpIsAppreciated]

Calculate the dubbelintegral (y/x) dxdy over D (i dont know how to write integral symbols here, sorry!) where D is given by
D: 3<=3x+3y<=5 and 5x<=y<=9x.

I have a few questions:
1. Does it make more sense to integrate y or x first and why?
2. How do i combine the two equations in D to get the proper limits for x and y.

i tried a few times but didnt get the right answer.
According to wolfram the limits should be y=[1-x,9x] and x=(1/10,1/6].
Why is the lower limit taken from the first equation in D but the upper limit taken from the second equation?

Any guidance in general would be appreciated, i missed a few lectures because i was sick and just started with these types of problems.

 

[Deleted member 4993]

Calculate the dubbelintegral (y/x) dxdy over D (i dont know how to write integral symbols here, sorry!) where D is given by
D: 3<=3x+3y<=5 and 5x<=y<=9x.

I have a few questions:
1. Does it make more sense to integrate y or x first and why?
2. How do i combine the two equations in D to get the proper limits for x and y.

i tried a few times but didnt get the right answer.
According to wolfram the limits should be y=[1-x,9x] and x=(1/10,1/6].
Why is the lower limit taken from the first equation in D but the upper limit taken from the second equation?

Any guidance in general would be appreciated, i missed a few lectures because i was sick and just started with these types of problems.

If you plot D: 3<=3x+3y<=5 and 5x<=y<=9x → you plot the limits. Wolfram apparently is integrating w.r.t. 'x' first.

So plot or sketch (and shade) on same axes:

y >= 1 - x

y <= 5/3 - x

that was your first "inequation". Now plot (and shade) on same axes:

y >= 5x

y <= 9x

There should be 4 lines on your plot (and two more for x & y axes) - make it dark and reasonably big. Label the lines and the points of intersections. If you still have questions - please write back (including a photograph of your graph).

 

[AnyHelpIsAppreciated]

If you plot D: 3<=3x+3y<=5 and 5x<=y<=9x → you plot the limits. Wolfram apparently is integrating w.r.t. 'x' first.

So plot or sketch (and shade) on same axes:

y >= 1 - x

y <= 5/3 - x

that was your first "inequation". Now plot (and shade) on same axes:

y >= 5x

y <= 9x

There should be 4 lines on your plot (and two more for x & y axes) - make it dark and reasonably big. Label the lines and the points of intersections. If you still have questions - please write back (including a photograph of your graph).

I plotted the lines and found the intersections (attached a photo). So if x varies from 1/10 to 5/18 and y from 5/6 to 3/2 how do i use that information to solve this dubbelintegral?

 

[AnyHelpIsAppreciated]

I plotted the lines and found the intersections (attached a photo). So if x varies from 1/10 to 5/18 and y from 5/6 to 3/2 how do i use that information to solve this dubbelintegral?

Should the inner limits (first integral) be described with variables and the outer one with number so that it becomes R2->R ?
I tried using the inner limits of x to be 1-y<=x<=5/3-y and the outer integral with the limits 5/6<=y<=3/2 but matlab couldnt compute it...

 

[AnyHelpIsAppreciated]

I solved it, you had to rearange it to 1<=x+y<=5 and 5<=y/x<=9 and then set u=x+y and v =y/x and use jacobian etc...