### MAT 132 Fall 2004 --- Review for Midterm 1

*References are to Stewart, Calculus - Concepts and Contexts, 2nd Ed.*
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 390]. 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 392, Example 6 page 393]. Exercises 7, 24, 43.

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 397; 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, Exercise 6].

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 360]. 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 419]. Understand how to compute the Midpoint
approximation `M` and how to apply Simpson's Rule
`S = (T + 2M)/3`. Exercise 23.