Chapter 16.1 Introduction
There are undefinite integrals and definite integrals

Chapter 16.2 Undefinite integral F(t) of the function f(t)
Undefinite integral F(t) is often named an integral F(t).
16-1a
Fig. 16-1
The derivative of the antiderivative function F(t) is this same function f(t)
and vice versa- the antiderivative of the function f(t) is a function F(t).
16-2a
Fig. 16-2
Integration and differentiation are the inverse operations.

Chapter 16.3 Definite integral of the function x(t)
Chapter 16.3.1 Introduction
The control theory has more a brush with the definite integral than with the undefinite integral. The latter enables an easy definite integral calculation.

Chapter 16.3.2 Definite integral of the function x(t) as a surface area under the function x(t)

16-3
Fig. 16-3
Definite integral from t1 up to t2 is a surface  area S uder a function x(t), so it’s a particular number e.g.  S=27.13. Something begins in time  t1=0 in automatic control theory usually, so we assume that  x(t)=0 for t<0.

16-4
Fig. 16-4
This is Fig. 16-3 version when t1=0 and t2=t. Please note that Definite integral is a function y(t)  now, not a concrete number S!
And most important now.
The definite integral=Surface Area calculus method.

16-5a
Fig. 16-5
We wil test this theory .

Chapter 16.3.3 Definite integral of the function x(t) as step function x(t)=1
Why the step x(t)=1? Because this function is as simple as possible and its area S calculus method is easy.
The definite integral of the function x(t) is an integral unit output y(t).
Call Desktop/PID/03_calka/01_calka_ze_skoku_jednostkowego.zcos
16-6a
Fig. 16-6
Click “Start”
16-7
Fig. 16-7
The x(t)=1 and it’s easy to calculate y(t) as a area surface from t1=0 up to t2=t under the x(t)=1. This area is y(t)=1*t16-8a

Fig. 16-8
There is some analogy  to the Fig. 16-2, but not at all. Why?
x(t)=0 for t<0 and x(t)=1 for t>0 and not for all t as in classical mathematical analysis
– there is a definite integral, Fig. 16-2 shows a definite integral

Chapter 16.3.4  Definite integral of the function x(t) as ramp type function x(t)=0.2*t
The ramp type x(t) and it’s easy to calculate y(t) as a area triangel surface from t1=0 up to t2=t under the  x(t)=0.2*t.
Call Desktop/PID/03_calka/02_calka_z_pily.zcos
16-9a
Fig. 16-9
Input is a ramp type x(t)=0.2*t.
Click “Start”
16-10
Fig. 16-10
We will count definite integral of the x(t) as a triangel surface. We don’t use mathematical analysis! This definite integral is an integral unit output y(t) too! Test that (for example for t=8 secdefinite integral of the x(t) is the same as parabola y(t)!

Chapter 16.3.4  Definite integral of the function x(t) as “potentiometer slider swinging” 
Call Desktop/PID/03_calka/03_calka_suwak_oscyloskop.zcos
16-11a
FIg. 16-11
Function x(t) is hand operated now. Use the digital meter to set x(t)=0.
Click “Start”
16-12
Fig. 16-12
Set gently x(t):
– x(t)=+0.025
– x(t)=+0.05
– x(t)=+0.025
– x(t)=0
– x(t)=-0.025
– x(t)=-0.05
– x(t)=-0.025
– x(t)=0
– ….itd
You note:
– big +x(t)–>+big speed of the y(t)
big -x(t)–>big speed of the y(t)
x(t)=0–>y(t) steady

Chapter 16.4 Conclusions
16-13a
Fig. 16-13
Integrating unit symbols
16-14
Fig. 16-14
Output y(t) is a definite integral of the input x(t))
Input x(t) is a derivative of the output y(t)

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