## 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/x`^{p} 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/x`^{p} 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