When you change your order of integration, you have to change the limits. The way I've been taught is to draw a graph in order to find the new limits. Is there a formula to find the new limits of integration?
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A diagram of the region is really more helpful, since you would need to invert the functions that define the boundaries of the integration region in terms of the new variable of integration. This is not a big deal for a simple figure, like a triangle or quadrilateral, but a region with curvilinear "sides" may have defining functions that don't invert easily (or at all). Since any sorts of functions might be used for the boundaries, a general formula won't have a convenient description.
Sometimes the diagram will also tell us that one particular order of integration will be definitely preferable to the other (I'm talking about two dimensions here, but the same will apply for more dimensions as well.)