Chapter 1

# An Aerial View of the Area Problem

IN THIS CHAPTER

Measuring the area of shapes by using classical and analytic geometry

Using integration to frame the area problem

Approximating area using Riemann sums

Applying integration to more complex problems

Seeing how differential equations are related to integrals

Looking at sequences and series

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 relatively 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 based ...

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