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Understanding Integrals

  1. Mar 26, 2015 #1
    1. The problem statement, all variables and given/known data
    Can a single integral be used to solve a multi-variable equation, and can a triple integral be used to find the area under an y = f(x) curve?

    What I'm getting at is whether or not single, double and triple integrals must be integrated with respect to their corresponding number of variables. My textbook shows single ##\int f(x) dx ##, double ##\int \int f(x,y) dy dx ## and triple ## \int \int \int f(x, y, z) dy dx dz ## but it never shows ##\int f(x,y)## nor ## \int \int \int f(x, y) dy dx dz ##

    2. Relevant equations
    n/a

    3. The attempt at a solution
    This isn't exactly a homework question, but I just want to understand the concept of integrals better.
     
  2. jcsd
  3. Mar 26, 2015 #2

    Mark44

    Staff: Mentor

    "Solve" and "equation" aren't the right words here - you don't "solve" an integral, you evaluate it. And in place of "equation", "function" would be an appropriate choice. When you evaluate an iterated integral, in the inner integrations you are integrating a multi-variable function along one axis to, eventually, get down to an expression in one variable that you can integrate.
    I suspose you could, but it seems like a wasted effort. You can use a double integral to find the area under a curve, y = f(x). For example, these two integrals produce the same value:
    $$\int_0^1 x^2 dx$$
    and
    $$\int_0^1 \int_{y = 0}^{x^2}~1~dy~dx$$

    You could turn this into a triple integral to get a volume that is numerically equal to the area of the preceding integrals.
     
  4. Mar 26, 2015 #3
    Is this the difference between a function and an equation:
    equation ## 10 = 5 + x ## and ## 22 = x^2 + xy + z ##
    function ##f(x) = 5+ x ## and ## f(x, y, z) = x^2 + xy + z ##
     
  5. Mar 26, 2015 #4

    Mark44

    Staff: Mentor

    More or less, as far a functions are concerned. In your first function example f maps a number x in its domain to a number 5 + x in its range. f(x) represents the "output" value for an input x value.

    In your second function example, f maps an ordered triple (x, y, z) in R3 to the real number x2 + xy + z.
     
  6. Mar 26, 2015 #5
    I think technically you could evaluate those types of integrals, but you won't really get any thing with meaningful results.
     
  7. Mar 26, 2015 #6
    The single integral ##\int^{\infty}_{0}e^{-x^{2}}dx## can't be evaluated by finding an antiderivative but can still be worked out by introducing a second integral ##\int^{\infty}_{0}\int^{x}_{0}e^{-x^{2}}\,dy\,dx## I guess it still isn't quite the same thing as working out the single integral because you get a different answer. I'm just trying to think of an integral analogy with something I saw in concrete mathematics for doing finite sums.
     
  8. Mar 26, 2015 #7

    LCKurtz

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    Science Advisor
    Homework Helper
    Gold Member

    I think you mean ##\int_0^\infty\int_0^\infty e^{-x^2-y^2}~dydx##, which gives the square of that integral, and which is usually evaluated by changing to polar coordinates.
     
  9. Mar 26, 2015 #8
    I think that's what I meant also :-p
     
  10. Mar 27, 2015 #9

    HallsofIvy

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    Do you understand that [itex]\int_a^b \int_c^d f(x,y,z) dxdy[/itex] would be a function of z?
     
  11. Mar 27, 2015 #10
    yeah if you were to integrate a 3rd time it would be wrt z I suppose.
     
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