Iterated integral

In <a href="/facts/Multivariable_calculus/PEs5SgUF">multivariable calculus</a>, an iterated integral is the result of applying <a href="/facts/Integral/V7lyV997">integrals</a> to a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of <a href="/facts/Function_of_several_real_variables/cyujR2q2">more than one variable</a> (for example 
 
 
 
 f
 (
 x
 ,
 y
 )
 
 
 {\displaystyle f(x,y)}
 
 or 
 
 
 
 f
 (
 x
 ,
 y
 ,
 z
 )
 
 
 {\displaystyle f(x,y,z)}
 
) in such a way that each of the integrals considers some of the variables as given <a href="/facts/Constant_(mathematics)/SxlxTWFT">constants</a>. For example, the function 
 
 
 
 f
 (
 x
 ,
 y
 )
 
 
 {\displaystyle f(x,y)}
 
, if 
 
 
 
 y
 
 
 {\displaystyle y}
 
 is considered a given <a href="/facts/Parameter/zFvVFMv9">parameter</a>, can be integrated with respect to 
 
 
 
 x
 
 
 {\displaystyle x}
 
, 
 
 
 
 ∫
 f
 (
 x
 ,
 y
 )
 
 d
 x
 
 
 {\textstyle \int f(x,y)\,dx}
 
. The result is a function of 
 
 
 
 y
 
 
 {\displaystyle y}
 
 and therefore its integral can be considered. If this is done, the result is the iterated integral

∫
        
          (
          
            ∫
            f
            (
            x
            ,
            y
            )
            
            d
            x
          
          )
        
        
        d
        y
        .
      
    
    {\displaystyle \int \left(\int f(x,y)\,dx\right)\,dy.}

It is key for the notion of iterated integrals that this is different, in principle, from the <a href="/facts/Multiple_integral/Fvl3Bkam">multiple integral</a>

∬
        f
        (
        x
        ,
        y
        )
        
        d
        x
        
        d
        y
        .
      
    
    {\displaystyle \iint f(x,y)\,dx\,dy.}

In general, although these two can be different, <a href="/facts/Fubini%2527s_theorem/q4lhtS3s">Fubini's theorem</a> states that under specific conditions, they are equivalent. 
The alternative notation for iterated integrals

∫
        d
        y
        ∫
        d
        x
        
        f
        (
        x
        ,
        y
        )
      
    
    {\displaystyle \int dy\int dx\,f(x,y)}

is also used.
In the notation that uses parentheses, iterated integrals are computed following the <a href="/facts/Order_of_operations/sWsoTzbH">operational order</a> indicated by the parentheses starting from the most inner integral outside. In the alternative notation, writing 
 
 
 
 ∫
 d
 y
 
 ∫
 d
 x
 
 f
 (
 x
 ,
 y
 )
 
 
 {\textstyle \int dy\,\int dx\,f(x,y)}
 
, the innermost integrand is computed first.

Iterated integral open-in-new

Iterated integral