Changing variables in multiple integrals

Since the variable in a definite integral is just a dummy variable, its name can be freely changed, from x to y, in the first equality above. It is clearly necessary that the dummy variables have different names when they occur in a multiple integral. A double integral can also involve a nonseparable fimction fix, y). For well-behaved functions, the integrations can be performed in either order. Thus,... 

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