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 that's 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?).
$$\int_a^b \int_c^d f(x)g(y) dy dx= \int_a^b f(x)\left[\int_c^d g(y)dy\right]dx$$
because f(x) depends only on x and is treated like a constant in the "dy" integral. Of course, with constant limits of integeration, $$\int_c^d g(y)dy$$, is just a constant, not depending on x, and can be taken out of the "dx" integral:
$$\int_a^b \int_c^d f(x)g(y) dydx= \left[\int_a^b f(x)dx\right]\left[\int_c^d g(y)dy\right]$$.

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:
$$\int_a^b \int_{\phi(x)}^{\psi(x)} f(x)g(y)dy dx\ne \left[\int_a^b f(x)dx\right]\left[\int_{\phi(x)}^{\psi(x)}g(y)dy\right]$$
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...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.

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!