Quick question about multiple integrals

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Discussion Overview

The discussion revolves around the properties of double integrals, specifically the conditions under which a double integral can be expressed as the product of two single integrals. Participants explore whether this is a general rule, a special case, or a potential error, with references to Fubini's theorem and the implications of variable limits of integration.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant questions whether the ability to split a double integral into a product of single integrals is a general rule, a typo, or a special case.
  • Another participant suggests that this splitting is akin to how constants can be factored out in single-variable integration, noting that the functions must be independent of the variable of integration.
  • A third participant identifies this property as Fubini's theorem and explains the conditions under which it applies, emphasizing the importance of constant limits of integration.
  • Some participants express confusion regarding the implications of switching the order of integration and the role of limits in determining whether the integral can be split.
  • One participant reflects on their educational experience, mentioning gaps in their multivariable calculus course that may have contributed to their misunderstanding.
  • Another participant responds to perceived criticism about their learning, indicating that their questions are a part of the learning process.

Areas of Agreement / Disagreement

Participants generally agree on the relevance of Fubini's theorem but express differing levels of understanding and clarity regarding its application, particularly concerning the limits of integration. The discussion remains unresolved regarding the educational context and its impact on learning.

Contextual Notes

Some participants note that the limits of integration are crucial in determining whether the integral can be split, highlighting that variable limits complicate the situation. There is also mention of gaps in the educational curriculum that may affect comprehension.

mongoose
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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.
 
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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.
 
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.
 
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.
 
mongoose said:
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.
 
hey...maybe that's why I'm asking questions!...duh!
 

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