Quick question about multiple integrals

In summary, the conversation discusses the concept of splitting a double integral into the product of two single integrals. This is known as Fubini's theorem and is a fundamental part of multivariable calculus. However, it is important to note that this can only be done when the limits of integration are constant. The conversation also touches on the importance of thorough learning, despite any shortcomings in teaching.
  • #1
mongoose
28
0
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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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
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.
 
  • #6
hey...maybe that's why I'm asking questions!...duh!
 

1. What is a multiple integral?

A multiple integral is a type of integral that involves integrating a function over multiple variables. It is used to find the volume, area, or mass of a three-dimensional shape or region.

2. How is a multiple integral different from a single integral?

A single integral involves integrating a function over a one-dimensional interval, while a multiple integral involves integrating over a two- or three-dimensional region. Multiple integrals are also more complex and require different techniques to solve than single integrals.

3. What are the different types of multiple integrals?

There are two types of multiple integrals: double integrals and triple integrals. A double integral involves integrating a function over a two-dimensional region, while a triple integral involves integrating over a three-dimensional region.

4. What is the purpose of using multiple integrals?

Multiple integrals are used to find the volume, area, or mass of a three-dimensional shape or region. They are also used in physics, engineering, and other fields to solve problems involving multivariable functions.

5. What are some common techniques for solving multiple integrals?

Some common techniques for solving multiple integrals include using Fubini's theorem, changing the order of integration, and using polar, cylindrical, or spherical coordinates. These techniques can help simplify the integral and make it easier to evaluate.

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