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 ex sin x dx [Example 4, Exercise 13]. Know the "exotic" integrations by parts: ln x dx, arctan x dx [Examples 2, 5, Exercise 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. [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/x2 on page 425 and
1/x in Example 1]. Understand why 1/xp
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/x1/2 [Example 5, Exercise 25] and 1/x. Understand why 1/xp gives a convergent integral from 0 to 1 if p < 1 and a divergent integral (no limit) otherwise [Exercise 49]. Exercises 5,17.
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.