Math 161
Spring 2000
Exam 2
- (30 pt) Determine if
the following integrals converge or diverge:
a) 
b) 
c) 
d)
e)
- (20 pt) Consider the solid
formed when the region, R, bounded by
(
) and the 
is revolved about the 
a)
Find
the volume of the resulting solid.
b)
Use
the part a) to find the 
of the centroid of the region R (hint: theorem of Pappus).
You may use the fact that 
- (5 pt) Again, consider
the fact that
. Find the value of 
. - (15 pt) Assume that
is a function such that the value of 
for Simpson’s
rule is always 30 times the value of 
for the midpoint
rule.
a)
Find
such that the error bound for Simpson’s rule is less than or
equal to the error bound for the midpoint rule.
b)
Does
this necessarily mean that Simpson’s rule is better than the midpoint rule for
this choice of 
? (Why or why not?)
c)
How
small should the length of the interval [a,b] be so that Simpson’s rule with 
has a smaller error
bound than the midpoint rule for 
?
- (20 pt) Consider the
parabola
, (
).
a)
Find
the length of this curve from 
to 
b)
What
is the surface area obtained when this curve from 
to 
is revolved about the
?
- (20 pt) Consider the
following triangular region:
a)
Use
any method that you wish to find the centroid of the region (besides quoting a
formula).
b)
Use part a) to find the
volume obtained when the region is revolved about the line 
