CHAPTER 7 Applications of Integration to Geometry
Concepts and Skills
In this chapter, we will review using definite integrals to find areas and volumes; specifically- area under a curve,
- area between two curves,
- volumes of solids with known cross sections,
- and volumes of solids of revolution (using disks and washers).
We’ll also review related BC topics, including - arc length;
- arc lengths, areas, and volumes involving parametrically defined functions;
- and area and arc length for polar curves.
Also for BC Calculus students, we’ll review the topic of improper integrals, including - recognizing when an integral is improper
- and techniques for determining whether an improper integral converges or diverges.
A. AREA
To find an area, we
(1) draw a sketch of the given region and of a typical element;
(2) write the expression for the area of a typical rectangle; and
(3) set up the definite integral that is the limit of the Riemann sum of n areas as n → ∞.
FIGURE N7–1
If f (x) is nonnegative on [a,b], as in Figure N7–1, then f (xk) Δx can be regarded as the area of a
typical approximating rectangle, and the area bounded by the x-axis, the curve, and the vertical lines x
= a and x = b is given exactly by
See Questions 1,5, and 10 in the Practice Exercises at the end of this chapter.