Question: List five other iterated integrals that equals to -
To be honest, I don't even know how to start this problem...
The problem asks to change the order of interated integration. By Fubini's Theorem order in iterated integration is not important. Thus there are six other representations of an integral each having,
That is why there are 5 others cuz one is already given.
List five other iterated integrals that equals to:
I write in the limits "completely".
I start with the "outside" limits: .
That is, ranges from toCode:y | - - -1+ - - - - - :::::|::::::::: :::::|::::::::: :::::|::::::::: ------+---------- x |
The next limits are: .
That is, ranges between the line andCode:y | - - -1+ - - - * - | /:| | /:::| | /:::::| ------/-------+-- x | 1
Now lay this diagram "on the floor"; the z-axis goes straight up.Code:z | | | - - - - *---------- y *::* *:::::* *::::::::* * * * * * * * / x
I can't draw this next part . . . hope you can follow.
With no restriction on , we have a triangular prism
. . with the above triangle as a cross-section, extending up and down infinitely.
But the limits are: and
. . That is, ranges between the "floor" and the slanted plane
So our triangular prism is cut off at floor-level below
. . and the slanted plane above.
And that is the solid we are dealing with.
Now if we change the order of integration, the limits are changed.
With the first of the answers, we have:
We will try to describe the same solid using this order of limits.
Starting outside: goes from to
Then: goes from to
Then: goes from to
So the integral is: .
. . I hope this helps . . .
Thank you for your explaination, I worked the problem and I was managed to get the answers of dzdydx, dxdzdy, and dxdydz.
However, for dydxdz, dydzdx, and dydxdz, I still don't understand why are they the same solid as the given one, why is y=x, y=z?
As another way of solving these, I work with the inequalities describing the region of integration. Start with the original integralOriginally Posted by tttcomrader
The inequalities describing the region are
To get the you asked for, we work with and rearrange the inequalities. We need the outer limit for first. From the first and third inequalities, we have
so we can write
For and , we have from the second and third inequalities
so we have for
The result is the new set of inequalities
and the integral
In writing the new inequalities, you can use any implications from the original inequalities. To make sure you haven't left any essential implications out, check that any triple that satisfies either one set of inequalities also satisfies the other.