Problem
13: 1998-1999
Given f(x) = x2 + 7.
(a) Find the approximate area bounded by f(x) and the
x-axis on the interval [0,4] with n = 8 rectangles using the midpoints
of each subinterval (i.e. the midpoint sum, not the left or right hand sum).
(b) Find the average value of f(x).
Difficulty rating: 
Solution:
(a) We first need to find the width of the rectangles; as always,
we use (b-a)/n, which equals 1/2. Unlike before, we do not want to use
the heights of the rectangles given by either the left or right endpoints; instead,
we want to use the heights given by the midpoints of each interval. Thus,
the rectangle with edges as x = 0 and x = 1/2 will have height
defined by f(1/4); the next rectangle will have height defined by f(3/4),
etc. Thus, to find the approximate area, we use the formula:
.
After much calculating, we get (1/2)(98.5)=49.25 as the approximate
area. Note that the (1/2) on the outside of the above formula is the length
of each interval, factored out as it is in each. Note that the actual area
of the interval is:
(b) To find the guaranteed mean value,
f(c), of the function, we use the formula:
In case you were wondering, 148/3 is the same as 49.333,
which came from the end of part (a) above.
