### MAT 132 Spring 2006 Review - Chapter 5

*References are to Stewart, Single Variable Calculus - SBU Edition*
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 21, 25, 53.

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:
`ln x dx`, `arctan x dx` [Examples 2, 5, Exercises 21, 23].

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. [Figures 2, 3 page 413]
Understand how to compute the Midpoint
approximation `M` and how to apply Simpson's Rule
[Box, page 418, Example 4] 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 423 and
`1/x` in Example 1]. Understand why `1/x`^{p}
gives a convergent integral from 1 to infinity if `p > 1`
and a divergent integral (no limit) otherwise [Example 4].

B: Function goes to infinity at a finite value `a`.
Understand how to calculate an integral from `a` to
`b` as the limit, as `t` goes to `a`, of the
integral from `t` to `b` *if that limit exists.*
Fundamental examples `1/x`^{1/2} [See Example 5] and
`1/x`. Understand why `1/x`^{p}
gives a convergent integral from 0 to 1 if `p < 1`
and a divergent integral (no limit) otherwise. Exercises 23, 25.
Be able to apply the Comparison Test [Example 9]
Exercises 17, 19.