Expert: Paul Klarreich - 10/13/2009

Question
QUESTION: the double integral of x^3sin(y^3)dydx (the first integral is between 1 and 0 and the second integral is between 1 and x^2)

reverse the order of integration and evaluate the integral.

I dont understand why the order of integration needs to be reversed and im not sure how to approach this problem. any help would be awesome

ANSWER: Questioner:   Olivia
Category:  Calculus
Private:  no
 
Subject:  double integrals
Question:  the double integral of x^3sin(y^3)dydx (the first integral is between 1 and 0 and the second integral is between 1 and x^2) reverse the order of integration and evaluate the integral. I dont understand why the order of integration needs to be reversed and im not sure how to approach this problem. any help would be awesome
.................................
Hi, Olivia,
--------------------------------------------------------------------------------
WARNING: THIS DISCUSSION CONTAINS MATERIAL DIFFICULT TO VIEW ON CERTAIN COMPUTING SYSTEMS.  VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.
--------------------------------------------------------------------------------

Is this what you mean?

{1  {x^2
|   |     ..... dy dx
}0  }1

If so, then note:
The SECOND integral is from 0 to 1.  (Not sure if you mean 0 to 1 or  1 to 0.)  It's not the first -- it appears first, but gets done last.  It means:

{x=1  {y=x^2
|     |     ..... dy dx
}x=0  }y=1

And you are supposed to do this:

{x=1 [ {y=x^2          ]
|    [ |     ..... dy  ] dx
}x=0 [ }y=1          ]

The ....., of course, means  x^3 sin(y^3), so it is:

{x=1    [ {y=x^2          ]
|   x^3 [ |   sin(y^3)dy  ] dx
}x=0    [ }y=1          ]

So you do the inner integral first, pretending that x is a constant.  (That's why we can 'take it out' here.)  

When you are done with it, since you substitute for y, all the  y's disappear.
Then you do:

{x=1
|   x^3(Whatever function of x you got) dx
}x=0

Simple, right?  But good luck integrating  sin(y^3).  I don't know how to do it, either.

So YOU HAVE TO REVERSE THE ORDER OF INTEGRATION to get anywhere.

Now, then, do you want to know how?

Give it a try, and if needed, send me a followup.


---------- FOLLOW-UP ----------

QUESTION: sorry the intergral was:
{1  {1
|   |     x^3sin(y^3) dy dx
}0  }x^2

i just wanted to see if i am on the right track:
i reversed the order of integration getting:
{1  {sqrt(y)
|   |     x^3sin(y^3) dx dy
}0  }0

{1
|   (sqrt(y)^4sin(y^3))/4 dy
}0

then:
{1
|   (y^2sin(y^3))/4 dy
}0

Im not sure how to integrate this function so i dont know what to do next. did i make a mistake somewhere?

Answer
QUESTION: sorry the integral was:
{1  {1
|   |     x^3sin(y^3) dy dx
}0  }x^2

i just wanted to see if i am on the right track:
i reversed the order of integration getting:
{1  {sqrt(y)
|   |     x^3sin(y^3) dx dy
}0  }0

{1
|   (sqrt(y)^4sin(y^3))/4 dy
}0

then:
{1
|  (y^2sin(y^3))/4 dy
}0

Im not sure how to integrate this function so i dont know what to do next. did i make a mistake somewhere?
........................................................
Hi, Olivia,

You did fine.  That is the same integral I got.  But it really shouldn't be difficult.

How about this:

{1
|  (y^2sin(y^3))/4 dy   << your integral (and mine)
}0

Let  u = y^3,   du = 3y^2 dy,   so y^2 dy = du/3.

That will be:

{
|  sin(u)/12 du
}

I think you can handle it from there.