## MAT132 Review for Midterm I

5.5 Be able to carry out change of variables in an indefinite integral by "substitution." (Examples 1, 3, Exercises 7, 8, 11, 12). The cases u=ax, u=ax+b should be automatic, but see Examples 2, 4, Exercises 3, 4, 9. Be able to solve definite integrals by substitution. Either method mentioned on p. 397 is OK. Exercises 37, 38, 47, 48.

5.6 Be able to do "classic" integration-by-parts problems: polynomial x trigonometric, polynomial x exponential. (Examples 1, 3, Exercises 1-6). Be able to do the exponential x trigonometric ones by double integration-by-parts and regrouping: (Example 4, Exercises 13, 14). Know the standard exotic examples (Examples 2, 5, Exercises 16, 19, 21)

5.8 Be able to compute an integral numerically using left-hand sum. Be able to do this by hand for n=2, 3, 4 subdivisions. Understand that the left-hand sum overestimates a decreasing function, etc. Understand that the trapezoid rule overestimates a function that is concave up ("holds water"), etc. and that the midpoint rule underestimates a function that is concave up, etc. (Exercises 1, 2, 3, 4).

5.9 Be able to deal with integrals over infinite intervals (Examples 1-4, Exercises 3, 5, 6, 7, 8). Know the rule that 1/xp is divergent on 1, infinity if p = 1 or p < 1 and convergent otherwise. Be able to deal with integrals of discontinuous integrands (Examples 5, 6, 7, Exercises 23-26). Know the rule that 1/xp is divergent on 0, 1 if p = 1 or p > 1 and convergent otherwise.

Use the Chapter Review on pages 437-441 for further reviewing. Concept Check 8, 9, 10, 11. Exercises 9-26, 39, 40, 45-50.

6.1 Be able to calculate the area enclosed by two curves: find intersection points and compute appropriate integral. (Examples 2, 3, Exercises 5, 6, 7, 8). Be able to compute area enclosed by parametric curve (Example 7, Exercises 27, 31).

6.2 Be able to compute a volume by slicing: find appropriate axis, set up integral, evaluate (Examples 2, 3, 4, Exercises 1-4). Be able to use similar triangles and the Pythagorean Theorem to set up integrals for geometric shapes (Examples 5, 6, Exercises 18-22). Be able to use the "cylindrical shell" method (Example 7, Exercises 9, 10).

Use the Chapter Review on pages 496, 497 for further reviewing. Concept Check 1, 2, 3; Exercises 1-11.

October 4, 2000