February 1st, 2008
Math 1B
1. (a) Find all psuch that R∞
1xpdx converges.
(b) Find all psuch that R1
0xpdx converges.
2. Why is it important to notice when integrals are improper?
(a) Evaluate the integral
Z1
−1
dx
x2.
(b) Now evaluate the integral from part (a) where you “forget” that it
is improper. Explain how the results you get are different, and why
the answer to (b) doesn’t really make sense.
(c) Repeat parts (a) and (b) for
Z1
−1
dx
x2/3.
It turns out that failing to notice that an integral is improper will
only run you into trouble if the improper integral diverges, and the
integral computed without noticing improperness does not diverge.
But since plenty of improper integrals diverge, it is still important to
notice when they are improper!
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