How do I evaluate the iterated integral in my textbook?

[Dx]

Evaluate the iterated integral: integral 1 to 0 double integral square root(y)to y then f(x+y)dx dy.

I don't understand iterated integrals in my textbook and am clueless how to get started. In the previous chapter it converted ot uising reimann sums butthis next chapter is vague and therefore not the best examples to work from. Can anyone help me solve for this?
Thanks!
Dx

 

[HallsofIvy]

Is the problem really "integrate from 1 to 0" or is it from 0 to 1 (that's more common). If 0 is the lower limit on the integral then it is "from 0 to 1".

Of course, we can't integrate this until we know what f(x,y) is!

 

[Dx]

Originally posted by HallsofIvy
Is the problem really "integrate from 1 to 0" or is it from 0 to 1 (that's more common). If 0 is the lower limit on the integral then it is "from 0 to 1".

Of course, we can't integrate this until we know what f(x,y) is!

Yes, ur right. its from 0 lower limit to 1 upper. This is one of those that's throws me of because all the problem says is
[inte]upper 1 to lower 0 [inte][squ](y) uppper to y lower limit (x+y) dx dy

Im not given f(x+y)
Dx

 

[Mike2]

iteration usually means "doing it more than once" in any other context. I think the author probably means the more commonly used term "multi-integral".

In your example, I think he means from lower limit x=sqrt(y) to upper limit x=y, in which case one usually treats y as a constant and integrates with respect to (wrt) x, getting x*x/2 + yx. Then replacing x in this result with the upper limit of y and subtracting this from the result of replacing x with x*x/2. Then after doing that integrating, then integrate wrt to x and replacing the upper and lower limits as usual.

 

[HallsofIvy]

Mike2, how does one get "x*x/2+ xy" when one doesn't know the function? Dx said he was asked to f(x+y) but didn't know f. That's a peculiar form and you may be right that it really is just x+y.

Dx, could you check the problem and tell us exactly what it says?

 

[HallsofIvy]

Mike2: I don't think it is a matter of multi-integral being a "more common name". "Iterated integral" and "multi-integral" are conceptually different things.
The iterated integral requires that we have a specific coordinate system and integrate in a specific order i.e. integrate first with respect to y and then with respect to x.
The "multi-integral", we are given a function, f, defined at each point of a region in, say, the plane and integrate f dA where dA is the "differential of area".

One of the important parts of a multi-variable calculus is showing that, by choosing a coordinate system, a mult-integral can always be converted to an iterated integral (in fact that's pretty much what you HAVE to do in order to actually integrated it). In it's most general form, that's "Fubini's Theorem".

 

[Dx]

Originally posted by HallsofIvy
Mike2, how does one get "x*x/2+ xy" when one doesn't know the function? Dx said he was asked to f(x+y) but didn't know f. That's a peculiar form and you may be right that it really is just x+y.

Dx, could you check the problem and tell us exactly what it says?

Straight out of the text verbatium.

Evaluate the iterated integral: integral 1 to 0 integral square root(y)to y (x+y)dx dy.