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 1.
 This exercise emphasizes that the value of a definite integral
is found as the limit of Riemann sums. Use the function g defined
in the background section.
 (a)
 Run the commands given in the background section to obtain a
right sum and find its limit.
 (b)
 Follow the pattern of the sample commands to obtain a left sum
and find its limit.
 (c)
 Follow the pattern of the sample commands to obtain a middle
point sum and find its limit.
 (d)
 What theorem in the text guarantees that the limits in (a), (b)
and (c) are the same? What property of g makes it possible to
apply that theorem here? Comment on your answer.
 2.
 Use the int command to try to find antiderivatives for the
following functions. What important part is missing from the answers
that Maple provided?
 (a)

 (b)

 (c)

This problem shows that Maple cannot always find a ``nice''
antiderivative. The problem may not be with Maple; some functions do
not have usable antiderivatives.
 3.
 Use the int command to try to evaluate the following
definite integrals.
 (a)

 (b)

The answer is complicated looking. Use evalf to approximate it.
 (c)

Use evalf to approximate the odd looking answer.
 4.
 Recall that
 (a)
 Consider . Note that this has the form
with . Find the antiderivative by hand
(not by Maple).
 (b)
 Use Maple to evaluate .
 (c)
 Since the answers in (a) and (b) are not the same, who is wrong?
Can it be that both you and Maple are correct? Explain what is
happening.
 5.
 Consider on the interval
[1,2].
 (a)
 Use Maple to set up M_{6}. Use evalf to evaluate the
expression Maple gives.
 (b)
 If M_{6} is used to approximate , what
accuracy is guaranteed by the error bound formula? That is, what is
the greatest amount by which M_{6} might differ from
? (This part of the exercise can be done by hand
on a sheet attached at the back of the Maple pages.)
 (c)
 Use (a) and (b) to give an interval in which the number
must be contained.
 (d)
 In fact, how good an approximation is M_{6}? Use int to
get the exact value of . (You should not have done
this prior to now.) Compute
. How
does this number compare with your answer in (b)? Explain why the two
numbers differ.
Next: About this document ...
Up: Labs and Projects for
Previous: Definite Integrals
Christine Marie Bonini
4/5/1999