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Examples of how to integrate with respect to x or y for an area?
I am learning how to determine whether to integrate with respect to x or y for an area. I would like a few examples on this with step by step explanations on why you chose wrt x or wrt y (or both) where you use the formulas and identify the top/right and bottom/left integrals. Also, it would be nice to include formulas for at least one or two of them to illustrate how that helps you make your decision.
It'd also be nice to have a conclusion statement of the concept.