MAC 2233/001006 Business Calculus, Spring 2011
CRN: 12577–12582
Assistants
: V. Kumari, Y. Xi, D. Ozcan, M. Assad, T. Zerihun, A. Rai
On the Midpoint Rule for numerically estimating the value of a definite integral
Assume that
f
is continuous and nonnegative on the interval [
a,b
].
The definite integral is
equal to the area of the region “under” the graph of
y
=
f
(
x
) from
x
=
a
to
x
=
b
. We saw
in a previous example that if an antiderivative of
f
can be found, then the exact value of the
area (and of the definite integral) can be obtained.
However, not all functions
f
have easy antiderivatives. In such cases, we would use numerical
integration. There are many sophisticated methods for numerical integration. We shall consider
a very elementary method, called the Midpoint Rule, to estimate the value of the area of the
region bounded by
y
=
f
(
x
),
x
=
a
,
x
=
b
and
y
= 0.
So we would want to estimate the area of shaded region in the diagram below.
b
a
We can divide the interval [
a,b
] into
n
subintervals, and break the shaded region into
n
vertical
“strips” as shown in the diagram below.
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 Fall '08
 DANIELYAN
 Calculus, shaded region, Midpoint Rule, subinterval, n subintervals

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