## MAT126 Review for Midterm II

5.4 Understand basic manipulations of indefinite integrals
(from 5.3, *Properties 1-5* p.368,369, and from 5.4
*Comparison Properties 1-3* p.384). *Example 1, Exercise 4*).
Understand both statements of the Fundamental Theorem, p.388:
1-The derivative of a definite integral with respect to its upper
endpoint is the value of the integrand at that endpoint ( *Examples 2,4,
Exercises 7,11,12,14,15,17* -remember chain rule!); 2-Evaluation
of a definite integral by the difference of the values of an
antiderivative at the endpoints (review section 5.3). Understand
the principle of the proof of statement 1, as sketched on p.389:
the rate of increase of the area is equal to the height of the
function.
5.5 Practice as many substitutions as you can. You need to
be able to recognize a likely "u", and that comes with practice.
(*Examples 2,3,4* and the more difficult *Examples 5,6,7*
-each of those involves a standard "trick"; you need to know
these elementary tricks. *Exercises 7-32*, as many as you can do.
For definite integrals, remember that you have to EITHER
transform your limits of integration to be the corresponding
u-values, OR rewrite your u-antiderivative in terms
of the original variable before evaluating. *Exercises 37-52*,
as many as you can do.

5.6 Again, practice is essential: you need to be able to
recognize what is the "u" and what is the "dv". I recommend
using "Formula 2" (the "u,v" formulation). Once you have
chosen "u" and "dv", write down du and v -this requires a
preliminary integration!-. *Example 1*-obvious case;
*Example 2*-less obvious but you should know this "trick"
(*Exercises 4,7*).
*Examples 3,4* each give an important wrinkle in applying
"parts." Applying it several times if necessary (keep very
careful track of your signs!! - *Exercises 5,6,12*);
or applying it twice and
then solving for the integral (*Exercises 13,14*).
Be familiar with these maneuvers.
*Exercises 1-24*, as many as you can do.

5.8 Remember the relation between left sum `L`_{n}. right sum `R`_{n},
trapezoidal rule `T`_{n}, midpoint rule `M`_{n} and Simpson's rule `S`_{2n}, that is:
`T`_{n} = (L_{n}+R_{n})/2 and `S`_{2n} = (2M_{n}+_{n}T)/3 (*Example 4*).
This will help you derive the formula for Simpson's rule. p.422 which you
should know. Understand that `L`_{n} underestimates and
`R`_{n} overestimates if
`f` is increasing, and vice-versa if `f` is decreasing.
Understand that `M`_{n} underestimates and
`T`_{n} overestimates if `f` is concave-up,
and vice-versa if `f` is concave-down. *Exercises 5,7,8,21,23,24*.

Use the Chapter Review for further reviewing.

- Concept Check 7,8,9.
- True-False 5 through 8
- Exercises 10,16,20,31,32,39,40

April 1 2000 - no fooling.