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How to write an iterated integral?

Consider the double integral x*y^2 dA where R is the region bounded by y=x^2 and y=2x. 
 
a) Write an iterated integral with the order dy dx that is equal to the above integral.
b) Write an iterated integral with the order dx dy that is equal to the above integral.
 
Please show complete work. Thanks.
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1 Answer

For a)    the integral is  ∫ dy ∫ dx  1     with the following limits:
                 The lower limit of the outer integral is 0
                  The upper limit of the outer integral   is  4
                  The lower limit of the inner integral is   y/2
                  The upper limit of the inner integral is sqrt(y)
 
For b)   the integral is  ∫ dx  ∫ dy  1    with the following limits
                    The lower limit of the outer integral is 0
                    The upper  limit of the outer integral is  2
                     The lower limit of the inner integral is x2
                      The upper limit of the inner integral is 2x
 
Both integral evaluate to the same result which is 4/3
 

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