6.1 Areas between curves. Understand that if f > g on an interval [a,b] then the area between the graphs is the area under f minus the area under g [Example 1]. Know how to solve a "region enclosed" problem: locate the intersection points - these will be the limits of integration [Example 2]. Be able to set up the problem as a y-integral when appropriate [Example 5]. Exercises 7,11. Be able to calculate the area enclosed by a parametric curve [Example 6] Exercise 33.
6.2 Volumes. Understand how slicing reduces the calculation of volume to a calculation of area and an integration [Discussion on pages 448 and 449] and how to implement the calculation [sphere, Example 1]. Know how to set up the integral for the volume of a solid of revolution [Examples 2, 5]. Know how to apply slicing to set up the volume integral for other solids [Example 7 and Example 8]. Be able to use the "cylindrical shell" method when appropriate [Example 9]. Exercises 5, 9, 21, 23, 29, 49.
6.3 Arclength. Be able to set up the integral giving the length of the curve traced out by the point with coordinates (x(t), y(t)) between t = a and t = b [Formula 1, Example 1 p.463, Exercises 3, 5, 9]. Special case: be able to set up the integral giving the length of the graph of y = f(x) between x = a and x = b [Formula 2 p. 463, Example 2, Exercises 8,9]. Since these integrals are in general difficult or impossible to calculate analytically (using anti-differentiation) be able to apply approximation methods to estimate them [Exercise 15a - use n=4 if doing by hand].
6.4 Average value. Be able to calculate the average value of the function f(x) on the interval [a,b]: divide the integral by (b-a). [Box p. 468, Example 1, Exercises 5, 7].
6.5 Work. Understand that if the force is a constant F, and displaces
its application point from a to b, the work done is
the product W = F (b-a); and that if the force varies with
distance x as F(x) the problem is handled by slicing
the interval [a,b] into infinitesimal subintervals of length
dx from x to x + dx over which the force can
be considered constant. The displacement from x to x + dx
involves an infinitesimal amount of work dW = F(x)dx, and
these infinitesimal amounts of work are summed by the integral: the
W is the integral of dW from a to b.
Be able to to convert problems of this type into integrals.
[Examples 1 and 2 page 472, Exercises 1-6].
A different application of Calculus is to problems where the distance that points are moved varies with the parameter x. In the simplest examples (e.g. emptying a rectangular tank of water over the top) the force is uniform, but points at one end of the problem (the bottom) have to travel farther than those at the other end (the top; here x is the height). The method is to slice the problem perpendicular to the x-direction, and to let dW be work corresponding to the slice at height x and thickness dx. [Example 3, Exercises 9, 11, 15].
In more complicated examples, the size of the slice may also vary with x [Example 4, Exercise 17]. Or the force may also vary with x [Exercise 13].
Be able to apply the "slice and integrate" method to all these problems.
6.7 Probability. Be able to check whether a given function f(x) can be a probability density function [Examples 1,2 page 487, 488] and how to interpret the integral from a to b of f(x)dx as a probability [Exercises 1, 3, 5]. Be able to calculate the average value (= the mean) of the random variable described by a probability density function f(x) [Example 3 page 489, Exercise 6c]. Be able to work with probability density functions given by graphs [Exercise 6] and equations [Example 1]; and in particular with exponentially decreasing probability density functions [Example 2, Examples 3 and 4 pages 489, 490, Exercises 7,9].