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- The integral is not iterated integral. Will this fact prevent us from swapping the order of surface and volume integrals? Why? Why not?
Failure of Fubini's theorem for non-integrable functions[edit]
Fubini's theorem tells us that (for measurable functions on a product of σ-finite measure spaces) if the integral of the absolute value is finite, then the order of integration does not matter; if we integrate first with respect to x and then with respect to y, we get the same result as if we integrate first with respect to y and then with respect to x. The assumption that the integral of the absolute value is finite is "Lebesgue integrability", and without it the two repeated integrals can have different values.
A simple example to show that the repeated integrals can be different in general is to take the two measure spaces to be the positive integers, and to take the function f(x,y) to be 1 if x=y, −1 if x=y+1, and 0 otherwise. Then the two repeated integrals have different values 0 and 1.
Another example is as follows for the function
The iterated integrals
and
have different values. The corresponding double integral does not converge absolutely (in other words the integral of the absolute value is not finite):
One like this: ##\int_V f(x, y, z)dV##What do you mean by "not iterated integral"?
I don't see a difference. What is a "not iterated integral"? Is it simple versus multiple? ##\int_V f\,dV## and ##\int_{V_z}\int_{V_y}\int_{V_x} f\, dx\, dy\, dz ## is the same thing.One like this: ##\int_V f(x, y, z)dV##
This can also be written as ##\iiint_V f(x, y, z)dV##. I've seen both styles in textbooks.
Here is an example of an iterated integral:
##\int_{y=0}^\pi\int_{x = 0}^\pi \sin(x + y)dx~dy##
One that is not an iterated integral...What is a "not iterated integral"?
Maybe, or maybe not. A double integral such as ##\int_R f\,dA## (where R is the region in the plane over which integration is to be performed) could be rewritten as two different iterated integrals: one in Cartesian form or one in polar form.Is it simple versus multiple? ##\int_V f\,dV## and ##\int_{V_z}\int_{V_y}\int_{V_x} f\, dx\, dy\, dz ## is the same thing.
I disagree. No calculus textbook that I've ever seen would write an integral like this: ##\int dx\,dy\,dz##.I would call this multiple or nested. It's not really an iteration. So the answer to my question is: A not iterated integral is a single integral. That means ##\int dV## versus ##\int dx\,dy\,dz## is more a linguistic issue than a mathematical.
Yes, I forgot to triple the integration symbol.I disagree. No calculus textbook that I've ever seen would write an integral like this: ##\int dx\,dy\,dz##.
Yes, but how is this not purely notational? To distinguish it linguistically appears hair splitting to me.Again, it's the difference between this triple integral ##\int_D f(x, y, z) dV## or ##\iiint_D f(x, y, z) dV## (not iterated) and this iterated integral ##\int_{z = z_1}^{z_2}\int_{y = y_1}^{y_2}\int_{x = x_1}^{x_2} f(x, y, z) dx~dy~dz##.
Because the iterated form contains information about the order in which integration is to be performed, in addition to possibly distinguishing between Cartesian coordinates or polar/cylindrical coordinates are to be used.Yes, but how is this not purely notational?