Double Integral: Order of Integration | 37

[Devilwhy]

Homework Statement

i am done with this question and get the answer 37
but it required us to change the order of integration

Homework Equations

The Attempt at a Solution

i have no idea...
i tried to do something like

[/B]
or

but none of them can obtain the answer 37

 

[andrewkirk]

A good way to do this sort of problem is to first draw the region of integration on the number plane.
Then, by rotating the page 90 degrees you can see what your limits of integration need to be for x when your outside variable of integration is y.
In this case, you will see when you draw and rotate the picture that you need to split the inner integral into three separate integrals.

 

[Devilwhy]

A good way to do this sort of problem is to first draw the region of integration on the number plane.
Then, by rotating the page 90 degrees you can see what your limits of integration need to be for x when your outside variable of integration is y.
In this case, you will see when you draw and rotate the picture that you need to split the inner integral into three separate integrals.

i did draw the region and rotate

but how can i do the integration?
i don't know how to deal with the inner integration now...

 

[nrqed]

i did draw the region and rotate

but how can i do the integration?
i don't know how to deal with the inner integration now...

Find the four relevant coordinates where the curves intersect (these are the four vertices of your black region). Then, break up the integration into three pieces (from the first point to the second, the second to the third and the third to the fourth). In each section determine what are the upper and lower limits of the black region. (for example, in the first region, you will integrate from the parabola to the upper straight line).

 

[geoffrey159]

The domain you should get by exchanging the double integral is ${\cal D } = \{ -2 \le y \le 4, \max (1,-y, \sqrt{ \max( 0,y) } ) \le x \le 2 \}$.
Have fun for the computations :-)

 

[geoffrey159]

I don't see the point of such exercise beside the ability to juggle with domains of integration.
The OP will likely be quite tired after all the computations (errors, checking, new errors, re checking ...), but I'm not sure I understand what it teaches.
He found 37 the simple way, why asking him to find 37 the hard way ?

 

[LCKurtz]

I don't see the point of such exercise beside the ability to juggle with domains of integration.
The OP will likely be quite tired after all the computations (errors, checking, new errors, re checking ...), but I'm not sure I understand what it teaches.
He found 37 the simple way, why asking him to find 37 the hard way ?

Well, I agree that the numerical answer isn't interesting. But learning how to switch the order is worthwhile. I would have had such a problem state to just set up the integrals without performing them. Some integrals are easier one way than the other and any insight that can be gained by practicing this kind of problem will likely be helpful when 3D material is addressed.

 

[Mark44]

I don't see the point of such exercise beside the ability to juggle with domains of integration.

Some integrals are easier one way than the other and any insight that can be gained by practicing this kind of problem will likely be helpful when 3D material is addressed.

I agree. Being able to switch the order of integration can sometimes make the difference between being able to do an integration versus not being able to do it at all.

 

[geoffrey159]

I agree with you that it is worthwhile to know how to switch the order of integration, but this exercise lacks structure in my opinion. It asks to follow the most difficult and lengthy path to a solution. Logically, the OP should get the best grade for his easy solution, and an F for all the others who followed the instructions like a herd of sheeps :-)

 

[nrqed]

I agree with you that it is worthwhile to know how to switch the order of integration, but this exercise lacks structure in my opinion. It asks to follow the most difficult and lengthy path to a solution. Logically, the OP should get the best grade for his easy solution, and an F for all the others who followed the instructions like a herd of sheeps :-)

You seem to think that the goal of doing problems is to get an answer. The final answer does not matter, what is important is understanding the process. And if someone can do a problem in more than one way, that person has a deeper understanding, even if one way is easier than the others. So I would definitely give more points to the "herd of sheep" who can figure out how to do it two ways.