# Thread: Question on multi-variable integration: f(x,y)=3*x*y

1. ## Question on multi-variable integration: f(x,y)=3*x*y

I'm trying to understand why I get the same answer when I perform a double integral of a simple function of x and y over specific ranges.

The function is f(x,y)=3*x*y.

In case one, I first integrate f(x,y) from 0 to x with respect to y. I then integrate that result from 0 to 3 with respect to x.

. . . . .$\displaystyle \int_0^3\, \int_0^x\, 3xy\, dy\, dx$

In the second case, I first integrate f(x,y) from y to 3 with respect to x. I then integrate that result from 0 to 3 with respect to y.

. . . . .$\displaystyle \int_0^3\, \int_y^3\, 3xy\, dx\, dy$

The answer is the same in both cases (30.375). Can anyone help me with an insight into a conceptual understanding of why that is?

2. Originally Posted by NYmathguy
I'm trying to understand why I get the same answer when I perform a double integral of a simple function of x and y over specific ranges.

The function is f(x,y)=3*x*y.

In case one, I first integrate f(x,y) from 0 to x with respect to y. I then integrate that result from 0 to 3 with respect to x.

. . . . .$\displaystyle \int_0^3\, \int_0^x\, 3xy\, dy\, dx$

In the second case, I first integrate f(x,y) from y to 3 with respect to x. I then integrate that result from 0 to 3 with respect to y.

. . . . .$\displaystyle \int_0^3\, \int_y^3\, 3xy\, dx\, dy$

The answer is the same in both cases (30.375). Can anyone help me with an insight into a conceptual understanding of why that is?
Why would you expect the answers to be different? Shouldn't the volume remain the same, regardless of the "direction" in which you first looked at it?

3. Originally Posted by stapel
Why would you expect the answers to be different? Shouldn't the volume remain the same, regardless of the "direction" in which you first looked at it?
Yes, the volume would be the same, but I'm looking for an understanding of why the limits of integration change. In the first case, the inner integral with respect to y goes from 0 to x, but in the second case, the inner integral with respect to x goes from y to 3 in order to get the same result. Why? If all we're doing is changing the order that we integrate over a surface, why does the inner integral with respect to x not go from 0 to y instead?

4. Usually, a good surface drawing will clear this up. Are you painting the same area on the x-y plane? Draw representative rectangles in the x- and y- directions.

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