Math 151-copyright Joe Kahlig, 13C
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Sections 6.2: Area
Computing the area of a region is easy when you can reduce the region to some geometric figures:
rectangles, squares, triangles, or trapeziods. For the figure to the right, computing the exact area by
just geometric figures is not possible since one side of the region is not a line. To compute this area
we need the following information.
Definition: The numbers a = x0 , x1 , x2 ,..., xn = b
where a = x0 < x1 < x2 < ... < xn = b is 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 }ni=1
The location in each subinterval where we compute
the height is denoted by x∗i .
Example: Use the partition P = {1, 1.2, 2, 2.5, 4} for the interval [1, 4] along with the function
f (x) = x3 + 2 to answer the following.
A) Find ||P ||.
B) Find the sum of the approximating rectangles if x∗i is the left endpoint of the subintervals. Sketch
the rectangles used on the graph.
Math 151-copyright Joe Kahlig, 13C
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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.
Math 151-copyright Joe Kahlig, 13C
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General form for the sum of the approximating rectangles:
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?
Example: Approximate the area under the function f (x) = x2 + 3 on the interval [1, 7] using a partition that has equal subintervals.
A) L2 =
B) R3 =
Math 151-copyright Joe Kahlig, 13C
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C) Rn =
D) Compute the actual area.
General form for x∗i on the interval [a, b]
The following formulas all assume that the interval [a, b] is partitioned into equal sub-intervals, i.e.
every rectangle has the same base. This means that ∆x = b−a
n where n is the number of rectangles used.
Right: x∗i = a + i∆x
Left: x∗i = a + (i − 1)∆x
Midpoint: x∗i = a +
(2i − 1)∆x
2
Note: When computing the actual area using Riemann sums, we usually use the right sum(right
Math 151-copyright Joe Kahlig, 13C
endpoint) since this method has the easiest formula.
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Math 151-copyright Joe Kahlig, 13C
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.
f (x) = x2 + 7x
Example: The following represents the area under a function f (x) on an interval [a, b]. Find f (x), a,
and b. Assume that a right endpoint was used when setting up the summation. Note: There may be
more than one right answer.
n
3 X
A) lim
n→∞ n
i=1
n
10 X
n→∞ n
i=1
B) lim
s
1+
3i
n
1
10i
1+ 7+
n
3