Chapter 2: The Denite Integral: Denition
13
and an upper estimate of
(2.2)
(14.5 2) + (22 2) + (27.5 2) + (31 2) + (32.5 2) = 255 m.
The gap between the two estimates is now much smaller, with a maxim
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MATH 1003 Integral Calculus and Modelling
we have x = (b a)/N. If a > b we can use the same formula. The only new feature
is that now x is negative. From an algebraic point of view this has no eect
6
MATH 1003 Integral Calculus and Modelling
In the special case where F (x, t) = kx for some constant k > 0 this reduces to the
equation of simple harmonic motion:
(1.7)
m
d2 x
= kx.
dt2
This is an ex
24
MATH 1003 Integral Calculus and Modelling
Note that in this example none of the three Riemann sums give the lower or upper
Riemann sum. The maximum value of sin x occurs at x = /2, which is inside
8
MATH 1003 Integral Calculus and Modelling
Innite Sequences (Stewart 12.12)
At several points in the course we have to deal with innite sequences of numbers.
We conclude this introduction with a shor
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MATH 1003 Integral Calculus and Modelling
a
x
b
Figure 1
The middle expression here is an example of a (general) Riemann Sum for f (x)
on the interval [a, b]. Its value obviously depends on the cho
Chapter 2: The Denite Integral: Denition
17
The number LN is called a Riemann Lower Sum for the function f on the interval [a, b]. It depends not only on N, but also on f and the interval [a, b]. Simi
Chapter 2: The Denite Integral: Denition
15
velocity m/sec.
30
20
10
0
Dierence in Area
0
2
4
6
8
10
time t
Figure 3
In mathematical terms this means that both the upper and lower estimates approach a
2
MATH 1003 Integral Calculus and Modelling
A Note on Notation
In most of the examples discussed here the independent variable is taken to be x or t.
Often t will stand for time, and the derivative of
4
MATH 1003 Integral Calculus and Modelling
The big dierence is that, for the falling body, the time dependence was given by
a simple formula f (t) = gt, but in this example there is no such formula.