quick question about multiple integrals

mongoose

i was looking through a book and came across a double integral that was split into the product of two single integrals.

it was int (x^n)(y^n ) dxdy split into (int x^n dx)(int y^n dy)

i just finished a course in multivariable calculus(it was by no means thorough), and i didn't know that you could do this.

is this a general rule? a typo? a special case?

thanks in advance.

ice109

i think thats just like how a constant comes out of the integrand in single variable integration. since you can treat x^n as a constant with respect to int(y^n*x^n)dy and vice versa you can break it up. of course though you can't brake up integrals over products of the functions of the variable you're integrating over.

HallsofIvy

That's "Fubini's theorem" and I would have thought it would be a fundamental part of any multi-variable calculus class (sure you didn't nod off during that class?).
[tex]\int_a^b \int_c^d f(x)g(y) dy dx= \int_a^b f(x)\left[\int_c^d g(y)dy\right]dx[/tex]
because f(x) depends only on x and is treated like a constant in the "dy" integral. Of course, with constant limits of integeration, [tex]\int_c^d g(y)dy[/tex], is just a constant, not depending on x, and can be taken out of the "dx" integral:
[tex]\int_a^b \int_c^d f(x)g(y) dydx= \left[\int_a^b f(x)dx\right]\left[\int_c^d g(y)dy\right][/tex].

Notice that I added the constant limits of integration which you did not have in your integral: that's important. If the limits of integration on the "dy" integral depend on x, you cannot do that:
[tex]\int_a^b \int_{\phi(x)}^{\psi(x)} f(x)g(y)dy dx\ne \left[\int_a^b f(x)dx\right]\left[\int_{\phi(x)}^{\psi(x)}g(y)dy\right][/tex]
since the expression on the left would be a number while the expression on the right will be a function of x.

mongoose

no...didn't nod off in class...it's community college, they even skipped the whole section on infinite series...go figure

but thanks, that makes sense now. i just didn't get that it's a consequence of switching the order of integration. it wasn't immediately obvious to me.

ice109

no offense, really don't be offended, but ****ty teachers is no excuse for not learning a subject thoroughly. and you'll do well if you remember this.

mongoose

hey...maybe that's why i'm asking questions!...duh!