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 maximum possible
error of 255 200 = 55 m. It is very instruc
26
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 on our
formulas. Geometrically it means that in the Ri
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 example of a second order dierential equation, since it i
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 the
range of integration. Up to this point the function
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 short summary of the basic facts about such
sequences.
Inni
22
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 choice of the ci . By taking N
large enough, we can make L
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]. Similarly, let
UN be the total area of the larger rectangle
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 common limit as the size of the steps shrinks towards
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 any quantity depending on t can be
thought of as the r
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. The function
f is specied by its graph, rather than alg