Tuesday, January 17
−
Lecture 8 :
Numerical integration
.
(Refers to section 5.3)
After having practiced the problems associated to the concepts of this lecture the student should
be able to
:
Apply the Trapezoidal and the Simpson's rule to approximate the value of a definite
integral.
8.1
Recall
−
Recall the statement of the
Second fundamental theorem of calculus
:
If
f
is continuous on the closed interval [
a
,
b
] then the function
F
(
x
) defined as
is continuous, differentiable and
F
′
(
x
) =
f
(
x
). Hence
It says that for any continuous function
f
(
t
) on an interval [
a
,
b
] there exists an
antiderivative defined on this interval.
8.2
Definition
−
Functions which can be expressed as sums, powers, products or
quotients of polynomials, trig functions, exponential functions and their inverses are
called
elementary functions
. These are essentially all the functions which have been
studied up to now in highschool. One may wonder if there exist functions which are not
elementary functions. If so how can we represent them?
Mathematicians have tried to find the integral
for a long time without success. Since its integrand is a continuous function on the
real line this integrand must have an antiderivative.
It was eventually shown that this integral is not an elementary function. And so we
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 Winter '08
 ZHOU
 Calculus, Approximation, Derivative, nmidpoint approximation

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