Changing Order of Integration for Multiple Integrals  (example continued)

 

 

The next step is to draw the projections of the intersecting surfaces on to the

xy and xz planes as shown below to obtain the limits of integration.

                                     

 

So      0  ≤  y  ≤   1 – x       for the first integration on the variable,  y

 

 

 

                                    

 

 

So    0  ≤  x  ≤    √ ( 1 – z )    for the second integration on the variable,  x

 

 

 

The remaining integration is on the third variable, z.    In this case  0  ≤  z  ≤  1.

 

 

                     z = 1   z = √ ( 1 – z )    y = 1 - x

           I    =     ∫           ∫                        ∫       f(x,y,z)  dy dx dz          (result)

                     z = 0   x = 0                 y = 0

 

 

Click here for another example.

 



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