6
ApPLICATIONS OF
D EFINITE INTEGRALS
OVERVIEW I n Chapter 5 we saw that a continuous function over a closed interval has a
definite integral, which is the limit o f any Riemann sum for the function. We proved that
we could evaluate definite integrals usi
S ECOND-ORDER
DIFFERENTIAL EQUATIONS
OVERVIEW I n this chapter we extend oUI s tudy o f differential equations t o t hose o f second
order. Second-order differential equations arise i n m any applications i n t he sciences and
engineering. For instance, t
16
INTEGRATION IN
VECTOR FIELDS
OVERVIEW I n this chapter we extend the theory o f integration to curves and surfaces in
space. The resulting theory o fline a nd surface integrals gives powerful mathematical tools
for science and engineering. Line integra
15
MULTIPLE INTEGRALS
OVERVIEW In this chapter we consider the integral o f a function o f two variables f (x, y )
over a region in the plane and the integral o f a function o f three variables f (x, y , z) over
a region in space. These m ultiple integral
7
TRANSCENDENTAL
FUNCTIONS
OVERVIEW Functions can be classified into two broad complementary groups called
algebraic functions a nd transcendental functions (see Section 1.1). Except for the trigonometric functions, our studies so far have concentrated on
5
INTEGRATION
OVERVIEW A great achievement o f classical geometry was obtaining formulas for the
areas and volumes o f triangles, spheres, and cones. I n this chapter we develop a method to
calculate the areas and volumes o f very general shapes. This met
4
ApPLICATIONS OF
DERIVATIVES
OVERVIEW In this chapter we use derivatives to find extreme values o f functions, to
determine and analyze the shapes o f graphs, and to find numerically where a function
equals zero. We also introduce the idea o f recovering
3
DIFFERENTIATION
OVERVIEW I n the beginning o f Chapter 2 we discussed how to determine the slope o f a
curve a t a p oint and how to measure the rate a t which a function changes. Now that we
have studied limits, we can define these ideas precisely and
2
LIMITS AND CONTINUITY
OVERVIEW Mathematicians o f the seventeenth century were keenly interested in the study
o f motion for objects on or near the earth and the motion o f planets and stars. This study
involved both the speed o f the object and its dir
1
FUNCTIONS
OVERVIEW Functions are fundamental to the study o f calculus. In this chapter we review
what functions are and how they are pictured as graphs, how they are combined and transformed, and ways they can be classified. We review the trigonometric