Review Sheet for First Hour Exam - Calculus II

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The following is a list of questions on some of the topics we've covered
in this course so far.  Some of this material will also be covered on the
exam.  Material not mentioned here may also be covered on the exam.

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1.  How do you use Riemann sums to approximate the area under a graph on a
bounded interval?  Give an example.

2.  State the Fundamental Theorem of Calculus (FTOC).  What is the
geometric interpretation of the FTOC?  What relationship between the
definite integral and the derivative function is being expressed by the
FTOC?  Explain this relationship clearly.

3.  What is the difference between a definite integral and an indefinite
integral?  Explain the relationship clearly.

4.  What are the algebraic properties of integral, that is, those relating
to a sum of integrals and to a constant multiple of integrals?  What are
the integrals of the basic functions: polynomial and power functions; sine
and cosine functions; exponential functions?  

5.  What is the substitution technique for integration?  With which
differentiation rule is it associated?  Explain this association.  Give
several examples of integrals to which the substitution technique could be
applied.  What is inverse substitution?  Give several examples of
integrals to which trigonometric substitution could be applied.

6.  How do you change limits when using substitution?  Give several
examples.

7.  What is the integration-by-parts technique for integration?  With
which differentiation technique is it associated?  Explain this
association.  Give several examples of integrals to which integration-
by-parts could be applied.  What does LIATE mean and how do you use it?
What is a reduction formula?  Give an example of a reduction formula and
derive it.

8.  What is a rational function?  When is a proper rational function?  
Give an example.  What is a partial fraction decomposition?  Give an 
example.  How is a partial fraction decomposition used to evaluate an 
integral?  Give an example of an integral to which a partial fraction 
decomposition could be applied.  When is completing-the-square a useful 
technique for integration?  Give an example.

9.  Describe the various techniques for numerical integration --
left-hand approximation, right-hand approximation, midpoint rule,
trapezoidal rule, Simpson's Rule.  Explain the geometric rationale as well
as the algebraic formulation behind each one.  Give an example of each
one.

10.  Describe the error terms for the techniques for numerical
integration.  What is the dependence of each of the techniques on the
number of subintervals used?  On which derivative does the error term for
the techniques depend?
 
11.  What is an improper integral?  How many kinds are there?  Give an 
example of each kind.  How do you compute an improper integral?  Give 
examples.  

12.  How do you compute the area between curves?  Give examples of 
computing this area both dx and dy.  When do you use one over the other?  
Give an example.