Whitman College: David Guichard's "Calculus, Chapter 7: Integration, Section 7.1: Two Examples"
Read Section 7.1 (pages 145-149). This reading motivates the integral through two examples. The first addresses the question of how to determine the distance traveled based only on information about velocity. The second addresses the question of how to determine the area under the graph of a function. Surprisingly, these two questions are closely related to each other and to the derivative.