Chapter Introduction

Integration solves an ancient mathematical problem—finding the area of an irregular region.

The basic problem in integral calculus is finding the area under a curve. You may wonder why calculus deals with two seemingly unrelated topics: tangent lines on the one hand and areas on the other. One reason is that both are computed using limits. A deeper connection is revealed by the Fundamental Theorem of Calculus, discussed in Sections 5.3 and 5.4. This theorem expresses the “inverse” relationship between integration and differentiation. It plays a truly fundamental role in nearly all applications of calculus, both theoretical and practical.