Math 166
Summer 2002
Exam 2
- (18 pt) Evaluate the
following integrals:
a) 
b) 
where 
c)
- (15 pt) Suppose that
you have a function,
, such that the second derivative is opposite in sign
from the fourth derivative (
) and you wish to approximate 
where 
. Find the smallest value of 
for which the
error bound for Simpson’s rule is less than or equal to the error bound
for the midpoint rule. - (20 pt) Locate the
centroid of the region bounded by the
axis and the curve 
- (12 pt) Find the
value(s) of
for which the
total area under the curve 
is 1 (for this
problem 
). - (15 pt) A large rain
gutter is constructed with equilateral triangular cross-sectional ends of
length 2 feet. How much force must the end be able to withstand if water
weighs 62.5 pounds per cubic foot?
2
ft.
- (15 pt)Consider the
function
a)
Set
up the integral for the length of this curve for 
Evaluate it for 
b)
Use
this information to predict the value of 
c)
Set
up the integral for the surface area obtained when this curve (
) is revolved about the 
axis. Evaluate it for 
- (15 pt) Consider the
part of the circle
that lies above
the line 
Locate the
centroid of this region (hint: you may use symmetry to assume that the
centroid is located on the line 
and you may use
the fact that the volume of a sphere of radius 
is given by 
).