Introduction

8.1 Introduction

Elementary geometry gives us formulas for the area of figures with straight edges or circular boundaries — triangles, rectangles, trapeziums, circles. These formulas are essential to countless real-life applications of mathematics, but they only go so far: they cannot handle a region whose boundary is a general curve. To find the area enclosed by curves, we need the tools of Integral Calculus.

In the previous chapter, we saw that a definite integral can be interpreted geometrically — it represents the area bounded by the curve $y = f(x)$, the ordinates $x = a$ and $x = b$, and the $x$-axis, obtained by treating that area as the limit of a sum of thin rectangular strips. In this chapter, we build directly on that idea. We shall study a specific application of definite integrals: finding the area under simple curves, the area of regions bounded by lines and arcs of circles, parabolas, and ellipses (restricted to their standard forms), and — going further — the area of regions bounded by combinations of such curves.

What We Will Study

• Area under simple curves — the area between a curve, an axis, and a pair of bounding lines.
• Area between two curves — the region enclosed between two curves, found by combining the areas found above.
• Standard curves — circles, parabolas, and ellipses in their standard forms, together with lines.
• Regions bounded by combinations — for example, the area common to a line and a parabola, or a circle and a line.

Working out exactly how to set up and evaluate the integral for each of these situations is the subject of the sections that follow.