grading the exam: problems with setting up integrals

Aside from the usual difficulty in recognizing upper and lower limits of integration in theta,  one common confusion that people have is confusing the values of coordinates at the boundary of the region of integration--which need to become input into the selection of limits of integration--with the form of the integrand. For instance, if a cylinder has radius R, and the integrand has x in it, in the integrand x is written as R cos(theta) instead of r cos(theta) as it should be.