5.9 Understand the definition of an improper integral as a limit of definite integrals (Text on pp.427-428). Understand that the integral converges if the limit exists, and diverges otherwise (Definition 1 on p.428). Examples 1,2,3 are important. Understand the behavior of the 1/xp integrals from 1 to infinity: converge if p>1, diverge otherwise. Exercises 5,6,7,8 (be ready to use substitution!). Also Exercises 11,14,17,18 which use a preliminary integration by parts. Understand the behavior of the 1/xp integrals from 0 to 1: converge if p<1, diverge otherwise (Definition 3 on p.431). Examples 5,6,7. Understand Example 8 - requires integration by parts. Exercises 23,24,25,26. Understand that ln(x) gives a convergent integral from 0 to 1. Comparison Theorem and applications will not be covered in this examination.
6.1 Be able to do a problem like Example 2 (find area enclosed by two curves): locate intersection points, set up integral with correct sign (need to know which curve is higher), and evaluate Exercises 5,11,13,15. Be able to integrate with respect to y when appropriate (Example 5, Exercises 11,12. Be able to use numerical techniques (Example 4, Exercises 21-24). Be able to handle curves that intersect twice or more (Exercise 25). Areas enclosed by parametric curves will not be covered in this examination.
6.2 Understand how to calculate a volume by the method of slices: determine your axis of integration, find volume of infinitesimal slice (dV = A(x)dx if x is the variable of integration, and integrate dV between appropriate endpoints. See "Definition of Volume," p.457. For solids of revolution the axis of integration is the axis of symmetry (Examples 2,3,4, Exercises1-6). Understand how to set up integrals for more general volumes (Examples 5,6, Exercises 20,21,24,26,28). Be able to use the method of circular shells, especially when the slice method leads to difficult equations or integrals (Example 7, Exercises 41,42).
6.3 Be able to set up the integral giving the length of the graph of f(x) between x=a and x=b. Examples 2,3,4, Exercises 5,6,7.
6.4 Understand the definition of the average value of a function on an interval (Box p.470). Example 1, Exercises 5,11,12,13
6.5 Understand how the formula Work=Force x Distance becomes an integral when the force varies over the distance. Examples 1,2, Exercises 1,2,3,4 or when both force and distance vary during the problem Example 3, Exercises 7,11,13a,14
to come
Use the Chapter Review for further reviewing.
Concept check: 1-7.
Exercises 1-4 (area)
Exercises 5,8,9-12a (volumes)
Exercises 13,14 (arc length)
Exercises 16, 17a (work)