5.5 Substitution rule. Know how to recognize the "outer" function
`f` and the "inner" function `g` in the integrand
`f(g(x)) g'(x) dx`, and how to then simplify the integrand
by rewriting it as `f(u)du`. [Example 1 page 387]. Remember,
in an indefinite integral, to return to the original variable
(in this case, `x`) for your answer.
Know how to transform the limits of integration in a definite
integral [Equation 5 page 389, Example 6 page 390]. Exercises 7, 24, 41.

5.6 Integration by parts. Know how to choose `u` and `dv`
so that `v du` will be an easier integrand than `u dv`.
[Note at bottom of page 394; Exercises 3,5]. Know when to do two
consecutive integrations by parts [Example 3, Exercise 7]. Know the
method for treating integrands like `e ^{x} sin x dx`
[Example 4, Exercise 13]. Know the "exotic" integrations by parts:

5.9 Approximate integration. Know how to carry out a left endpoint
approximation `L` ("left-hand sum") for a definite integral, given
a number `n` of (equal) subintervals. Know also how to compute
the "right-hand sum" R [Example 2(a) page 357]. Know that if `f`
is increasing on an interval `[a,b]` then `L` underestimates,
and `R` overestimates, the integral of `f` from `a`
to `b`. Understand how to compute the Trapezoidal approximation
`T = (L + R)/2` and that `T` overestimates the integral if
`f` is concave up, and underestimates if `f` is concave
down. [Figure 5 page 415]. Understand how to compute the Midpoint
approximation `M` and how to apply Simpson's Rule
`S = (T + 2M)/3`. Exercise 25.

5.10 Improper integrals. A: Infinite interval of integration.
Understand how to calculate an integral from `a` to
infinity as the limit, as `T` goes to infinity, of the
integral from `a` to `T` *if that limit exists.*
[Fundamental examples `1/x ^{2}` on page 425 and

B: Function goes to infinity at a finite value

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.

6.2 Volumes. Understand how *slicing* reduces the
calculation of volume to a calculation of area and an integration
[Discussion on pages 449 and 450] 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, 25, 43.