

dr bob Supporting Diagrams When converting information about bounding curves to limits of integration in an iteration of a double integral, it is important to document the process of analyzing the information with annotated diagrams. "Shade the region of integration with equally spaced linear cross sections along the direction of the variable of integration, and label bullet point endpoints of a "typical" such linear cross section roughly in the middle with the starting and stopping equations of the variable of integration, with an arrow head midway along that cross section indicating the increasing direction of the corresponding integration variable." 

This sets up a parameterization of all the points in the region of integration which directly translates into the limits of integration.
Calculations involving continuous distributions of stuff (center of mass, centroid, probability) are all examples of the same mathematics: averaging a quantity (the stuff) over a 1, 2 or 3 dimensional region by integrating it against a weighting function which divides up the total using a weighting density function whose total integral is 1 (for all the stuff). The position vector in each case is averaged in this way to produce the desired measure of the stuff. Integrating with respect to the geometric measure (length, area, volume) of the region where the points are distributed in some subset of the whole space produces the geometric center called the centroid. For a distribution of mass or charge described by a continous density function, this produces the center of mass or charge. For a probability distribution this produces the expected value.
While centroids/centers of mass are not terribly interesting in themselves, they provide toy examples where students have some intuition of where the "center" roughly lies, and so can judge whether their computed value of the centroid makes sense, and then with a unequal weighting from an inhomogeneous distribution of mass, can have a sense of in which direction that distribution of mass will move the centroid. Thus they are good "toy" problems to learn how integration is used in applications.