Riemann Sums
Suppose that a function f is continuous and non-negative on an
interval [a,b].
Let's compute the area of the region R bounded above by the curve
y = f(x), below by the x-axis, and on the sides by the lines x = a and
x = b.
We will obtain this area as the limit of a sum of areas of rectangles
as follows:
First, we will divide the interval [a,b] into n subintervals
| [x0, x1],
[x1, x2], ¼,
[xn-1, xn] |
where a = x0 <
x1 <
¼ <
xn = b. (This is called a
partition of the interval.) The intervals need not
all be the same length, so call the lengths of the intervals
Dx1,
Dx2, ...,
Dxn, respectively. This
partition divides the region R into n strips.
Next, let's approximate each strip by a rectangle with height equal to
the height of the curve y = f(x) at some arbitrary point in the
subinterval. That is, for the first subinterval [x0,
x1], select some
x1* contained in that
subinterval and use f(x1*) as
the height of the first rectangle. The area of that rectangle is then
f(x1*)
Dx1.
Similarly, for each subinterval [xi-1, xi],
we will choose some xi*
and calculate the area of the corresponding rectangle to be
f(xi*)
Dxi.
The approximate area of the region R is then the sum
åin=
1 f(xi*)
Dxi of these rectangles.
Depending on what points we select for the
xi*, our estimate
may be too large or too small. For example, if we choose each
xi* to be the point
in its subinterval giving the maximum height, we will
overestimate the area of R. (This is called an upper
sum.)
If, on the other hand, we choose each
xi* to be the point
in its subinterval giving the mimimum height, we will
underestimate the area of R. (This is called a
lower sum.)
When the points xi*
are chosen randomly, the sum
åin= 1
f(xi*)
Dxi is called a Riemann
Sum
and will give an approximation for the area of R that is in between
the lower and upper sums. The upper and lower sums may be considered
specific Riemann sums.
As we decrease the widths of the rectangles, we expect to be able to
approximate the area of R better. In fact, as max
Dxi ® 0,
we get the exact area of R, which we denote by the definite integral
That is,
ó
õ |
b
a
|
f(x) dx = limmax
Dxi® 0 |
æ
ç
è |
n
å
i = 1
|
f(xi*)Dxi |
ö
÷
ø |
Notes
This definition of the definite integral still holds if f(x)
assumes both positive and negative values on [a, b]. If even holds
if f(x) has finitely many discontinuities but is bounded.
For a more rigorous treatment of Riemann sums, consult your
calculus text.
The following Exploration allows you to approximate the area under various
curves under the interval [0, 5]. You can create a partition of the interval
and view an upper sum, a lower sum, or another Riemann sum using that partition.
The Exploration will give you the exact area and calculate the area of your
approximation. To create a partition, choose which type of sum you would like
to see and click the mouse between the partition labels x0 and
x1.
Exploration
Key Concept
Let f be defined on [a, b] and let x0, x1,
¼, xn be a partition of [a. b].
For each [xi-1, xi], let xi* Î [xi-1, xi].
Then the definite integral of f over [a, b] as defined as
ó
õ |
b
a
|
f(x) dx = limmax Dxi® 0 |
æ
ç
è |
n
å
i = 1
|
f(xi*)Dxi |
ö
÷
ø |
