Integrals and shapes

One of my favorite essays on mathematics is Arnold's "On teaching mathematics." In it, he describes what he views as defects of the traditional treatments of real analysis and calculus in universities; and in particular, how the emphasis on rigor removed discussions of the beautiful geometry of Riemann surfaces and how they relate to calculus. As a specific example, he describes how one can tell which integrals are easy or hard to solve by looking at the genus of the associated Riemann surface. This week on Hidden Phenomena, you can learn what Arnold meant by this comment: what is the "genus of the Riemann surface associated to an integral," and how does it help you understand if the integral is easy or hard?