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
total work
`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].