Math 151
Section 6.2
Area
Approximate Area The numbers a = x0 , x1 , x2 ,..., xn = b
where a = x0 < x1 < x2 < ...< xn = b are said to be a
partition, denoted by P, of the interval [a, b].
The norm of the partition P, denoted ||P||, is the maximum
value of the length of the subintervals, !xi , in the partition
where !xi = xi " xi"1 , i.e., ||P|| = max {!xi }
n
i=1
The location in each subinterval where the height is
computed is denoted by x *i .
Example: Let the partition P = {1, 1.2, 2, 2.5, 4} on the interval [1, 4] with f ( x) = x 3 + 2 .
A. Find ||P||.
B. Find the sum of the approximating rectangles if x *i is the left endpoint of the subintervals. Sketch
the rectangles on the graph.
Math 151
C. Find the sum of the approximating rectangles if x *i is the right endpoint of the subintervals.
Sketch the rectangles used on the graph.
D. Find the sum of the approximating rectangles if x *i is the midpoint of the subintervals. Sketch
the rectangles used on the graph.
Example: Assume that f (x) is a continuous function that is above the x-axis on the interval [a, b].
A. What condition would the function f (x) have to have so that the sum of the approximating
rectangles will be an underestimate?
B. What condition would the function f (x) have to have so that the sum of the approximating
rectangles will be an overestimate?
Math 151
n
General form for the sum of the approximating rectangles: A = lim # f ( x *i )"xi
P !0
i=1
Example: Approximate the area under the function f ( x) = x 2 + 3 on the interval [1, 7] using a
partition that has equal subintervals.
A. L2
B. R3
C. Rn
D. Compute the actual area.
Math 151
General form for x *i on the interval [a, b]
Right: x *i = a + i!x
Left: x *i = a + (i !1)"x
Example: Set up the Riemann sum that will give the area under the graph for f (x) on the interval
[0, 5] using a left endpoint.
Example: The following represent the area under a function f (x) on an interval [a, b]. Find f (x), a,
and b for each.
3 n
3i
1+
#
n!" n
n
i=1
A. lim
10 n
)
n!" n
i=1
B. lim
1
3
#
10i &(
(
1+ %%7 +
%$
n (('