Chapter 1

An Aerial View of the Area Problem

In This Chapter

arrow Measuring the area of shapes by using classical and analytic geometry

arrow Understanding integration as a solution to the area problem

arrow Building a formula for calculating definite integrals using Riemann sums

arrow Applying integration to the real world

arrow Considering sequences and series

arrow Looking ahead at some advanced math

Humans have been measuring the area of shapes for thousands of years. One practical use for this skill is measuring the area of a parcel of land. Measuring the area of a square or a rectangle is simple, so land tends to get divided into these shapes.

Discovering the area of a triangle, circle, or polygon is also easy, but as shapes get more unusual, measuring them gets harder. Although the Greeks were familiar with the conic sections — parabolas, ellipses, and hyperbolas — they couldn’t reliably measure shapes with edges ...

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