The Magic Circle
A method of obtaining the Area of a
sinus curve without the use of Intergral calculus. Instead I resort
to Pythagoras triangles and elementary geometry . This is how it came to be . I have a paper
that is square, lets say 20 cm * 20 cm. Then draw in a Diagonal. I
clip the square out with scissors, and have the Square area of paper
in my hand. Then I fold the paper into a cylinder. I then hold the
paper up against the light, and the Diagonal has now become a Sinus
curve !
From this I can extrapolate that the
Diameter of a paper cylinder, that directly translates into Amplitude or signal strenght on the Y axis . The Radius describes the wavepeak and wave bottom . Calculating the circumference of the
cylinder this would translate into the height of a square . The lenght in X axis, of the period of
360 degrees. Lenght translates directly into wavelenght, and also a
concept of time, or X dimension . The lenght of the Sinus curve line, is equal to the paper square Hypothenus or diagonal .
I prefere to use the Pi because a Sphere or a Circle, has a reduced dimensional articulation. You can
express both Carthesian X, Y and Z dimensions with one Radial value, and constant Pi
!
The surface of a sphere with pi = 4* Pi * R^2.
Take a look how the surface of a sphere is done with Carthesian intergrals .
The surface of a sphere with pi = 4* Pi * R^2.
Take a look how the surface of a sphere is done with Carthesian intergrals .
Or use use MatLab : (online) . http://www.wolframalpha.com/widgets/view.jsp?id=ec78b0700064223c5cd7898812ecffae
Plot the number into the calculator and see :
SIN (X)
Wavepeak = 1 unit tall
Wavepeak (north) = Amplitude = Radius
Radius_cylinder = 1
Diameter = 2R = 1+1
wavelenght 1/2 wave = 3.14
in mathlab online type : sin (x) . from X = 0 . to X = 3.14 .
wavelenght 1/2 wave = 3.14
in mathlab online type : sin (x) . from X = 0 . to X = 3.14 .
O_circumf cylinder = Pi * 2R = Pi * D=
3.14 * 2 = 6.28 cm circumference
A_Cylinder = O_circumf * lenght = 6.28
* 6.28 cm = 39.4 cm^2
A_triangle = Lenght * height / 2 = 6.28
* 6.28 / 2 = 19.7 cm ^2 (north and south wave)
A_curve (north) = A_triangle / pi^2 =
19.7 /pi^2 = 1.99 cm ^2
A_curve (north and south) = (A_triangle
/ pi^2) *2= (19.7 /pi^2) * 2 = 4 cm ^2
10 SIN (X)
wavepeak (north and south ) = Radius =
10
SIN (X+10)
Wave is phased 10 units Left ( West )
SIN (X-5)
Wave is phased 5 units Right ( East )
SIN (X)
wavelenght (360 degrees rotation) is
3.14 *2 units long or 2 Pi long.
SIN (X*Pi)
wavelenght (360 degrees rotation) is 2
units long .
SIN (X*Pi/2)
wavelenght is 4 grid units long . With each half wave 2 grid units long .
Half wave = 1 * 2 = 2
Full wave = (1*2) * 2 = 4 grid units
Half wave = 1 * 2 = 2
Full wave = (1*2) * 2 = 4 grid units
SIN (X*Pi*2)
wavelenght is 1 grid units long, with 2 halfwaves per 1 grid unit .
Half wave = ( 1 / 2 ) = 0.5 cm
Full wave = ( 1 / 2 ) * 2 = 0.5 cm
Example 1
Draw a waveform that express 230V at 50 hertz .
Then another with 2 Ampere, delayed 5 millisec
U peak = 230* sqrt2 = 325V. Then you measure 230V in the wall socket.
Amplitude * SIN * pi * Hz*2
U_wave = 325 SIN (X*pi*50*2)
I_wave = 2 SIN ((X+5E-3) * pi* 50*2)
To capacitate that Ciruit into phase is an entirely different matter than to manage a calculator !
Half wave = ( 1 / 2 ) = 0.5 cm
Full wave = ( 1 / 2 ) * 2 = 0.5 cm
Example 1
Draw a waveform that express 230V at 50 hertz .
Then another with 2 Ampere, delayed 5 millisec
U peak = 230* sqrt2 = 325V. Then you measure 230V in the wall socket.
Amplitude * SIN * pi * Hz*2
U_wave = 325 SIN (X*pi*50*2)
I_wave = 2 SIN ((X+5E-3) * pi* 50*2)
To capacitate that Ciruit into phase is an entirely different matter than to manage a calculator !
Example 2
5 SIN (X*pi)
Draw the curve : squares of 1 cm.
Wavepeak is +5 and -5 to baseline, making it 10 units tall. The lenght is X that
is 1 unit per half wave. The entire period is 2 units long. If X axis
is seconds, the wavlenght is 1 hertz, or 1 period per second. Radius
or Amplitude is 5 .
Translate: Sinus curve to Cylinder
Radius_Amplitude_wavepeak = 5
Diameter_cylinder = 2R = 5 * 2 = 10
O_circumf_cylinder = Pi * 2R = Pi * D =
3.14 * 10 = 31.4 cm
Translate : Cylinder = Square,
Circumference = height
A_cylinder (north)= A paper square =
O_circumf * Length = 31.4 * 1 = 31.4
A_cylinder (north and south )= A paper
square = O_circumf * Length = 31.4 * 2 = 62.82 cm^2
Translate : Square into Triangle
A_triangle = L (lenght) * H (height ) /
2 = 62.82 / 2 = 31.4 cm^2
A_curve ( north ) = A_triangle / pi^2 =
31.4 / pi^2 = 3.18 cm ^2
I must admit that the valus is half of
what is expected so I need to work some more on it . Maybe double the
diameter ...
A_curve (north and south) = Wavelenght * O_circumf_cylinder / Pi^2
A_curve (north and south) = 1cm wavelenght * 31.4 / Pi^2 = 3.18
Example 3
3 SIN (X*pi * 5)
Radius = 3 cm
wavelenght ( lambda ) = 5
Q: Find the area of both north and south curve with respect to baseline.
O_circumf_cylinder = Pi * R * 2 = 3.14 * 3cm* 2 = 18.84
A_Cyl = Lambda * O_Cyl = 5 cm * 18.84 = 94.24 cm^2
A_curve ( north and south) = Lambda * O_cyl / pi^2 = 94.24 / pi^2 = 9.54 cm^2
3 lines of code !
Sum from 0 to 5.
Example 4
2 SIN ( X* pi * 4)
the last number 4, means this. 4 half waves per 1 unit lenght.
to obtain the area of a Phase ( or 1 period ) .
Lenght of half a period = 1 period / 4 = 0.25 cm long northwave
wavlenght 1 phase = (1/4) * 2
Radius = 3 cm
lambda = ( 1/4 ) * 2
A_phase = lambda * O_cyl / pi^2
A_phase = lambda * pi*2R / pi^2
A_phase = lambda * 2R / pi
A_phase = (1/4)*2 * (pi * 2*3 ) / Pi^2 = 0.95 cm^2
A_phase = (1/4)*2 * (pi * 2R ) / Pi^2
A_phase = 0.5 * 6 / Pi = 0.95 cm^2
Final result :
Areal_phase = lambda * 2R / pi
The notation "lambda" works as a "limit", that defines the end of your function. you can put it as long as you want or short as you want. you accumulate the area of the waveperiods. I havent figured out how to move the start limit of the function. Maybe try the SIN (X+10) to phase the waveform sideways .
Example 5
Lets say youn want the area of the waveform for 1 sec duration
3 SIN ( X * Pi * 4)
R = 3
Limit_end = 4 * 1 sec
A_phase_1 sec = Lim * O_cyl / Pi^2
A_Phase_1 sec = 4 * Pi * 2 * R / Pi^2
A_Phase_1 sec = 4 * 2 * R / Pi
A_Phase_1 sec = 4 * 2 * 3 / Pi = 7.63 cm seconds (siffix?)
for 9 seconds
3 Sin ( X * Pi * 4)
R = 3
Lim = 4 * 9 sec = 36
A_phase_1 sec = Lim * O_cyl / Pi^2
A_Phase_1 sec = 36 * 2 * R / Pi
A_Phase_1 sec = 36 * 2 * 3 / Pi = 68.75 cm sec
I got problems when trying to incorperate c= lightspeed 3E8. I realized the X axis either had to be time in seconds or meter. Or else I would want 2 different measures on the X axis, one for seconds and one for meter. X_sec would display 50 periods per second. X_meter would display one waveform at the lenght of c/hz = 3E8 / 50hz = 6 000 000 meter per sec . huh, meter per sec ....
so this didnt work :
U = 230V
lambda wavelenght = c / hz = 3E8 / 50 hz
U Sin (X Pi * (c / hz )) = didnt compute ...
Example 6
more formal terms
f(x) = R SIN (x*pi (N_halfwaves))
f(x) = R SIN (x*pi (hz*2))
Amplitude = Radius
Limit_Lambda_one wavelenght = (1 / hz) * 2 = (1 / 50 hz) * 2 = 0.2 * 2 = 0.04 cm
Hz = 50 periods per second ( 1 sec is 1 grid unit )
Amplitude (Radius) = 5
Area_sinus = Lim 2PiR / Pi^2
Area_sinus = Lim 2R / Pi
Area_sinus_one period = (1/hz)*2 * 2R / Pi
Area_sinus_one period = (1/50hz) * 2 * 2 * 5cm / Pi
Area_sinus_one period = 0.02 * 20 / Pi
Area_sinus_one period = 0.04 / Pi = 0.127 cm^2
Sin_curve lenght = Sinus is converted into a cylinder, then opened into a square. The diagonal on the square is equal to the lenght of the Sinus curve .
Sin_curve lenght = SQRT ( (2PiR)^2 + (1/Hz*2) )
Sin_curve lenght = SQRT ( (2 Pi *5 cm )^2 + (1/50 Hz*2) )
Sin_curve lenght = SQRT ( (31.4)^2 + (0.04)^2 )
Sin_curve lenght = SQRT ( 985 + 0.0016 )
Sin_curve lenght = SQRT 985.0016 = 31.3 cm long
Example 7
f(x) = R sin (x * pi * N_halfwaves)
f(x) = 3 sin (x * pi * 0.5)
R = radius of cylinder or waveform amplitude = 3
N_halfwaves = halfwaves per grid units= 0.5
Lambda = N_halfwaves * 2 = 1 period of 360 degrees rotation .
Lambda can be replaced by Limit or LIM_end .
LIM_end = the intergral version of "to_X". The formula does not apply for "from_X" other than set to 0 .
f(x) = 3 sin (x * pi * 0.5)
Area Sin = 2 Pi R * (LIM_end) / pi^2
Q1 : I want 1 period or a complete wavelenght
1 wavelenght is 2 halfwaves .
LIM_end = 1 period = ( (Desired LIM_end) / N_halfwaves ) * 2 = N grid units
LIM_end = 1 period = ( (1 / 1) / 0.5 ) * 2 = 4 grid units
Area_sin 1 period = (2 * Pi * R ) * ( (1 / 1) / 0.5 ) * 2 / Pi^2
Area_sin 1 period = (6Pi) * 4 / Pi^2= 7.639437268 cm^2
Q2 I want half a period
LIM_end = ( (1 / 2) / 0.5) * 2= 2 grid units
Area_sin 1 period = (2 * Pi * R ) * ( (1 / 2) / 0.5 ) * 2 / Pi^2
Area_sin 1 period = (6Pi) * 2 / Pi^2= 3.81 cm^2
Q3 I want a quarter period
LIM_end = ( (1 / 4) / 0.5) * 2 = 1 grid units
Area_sin 1 period = (2 * Pi * R ) * ( (1 / 4) / 0.5 ) * 2 / Pi^2
Area_sin 1 period = (6Pi) * 1 / Pi^2= 1.90 cm^2
Q4 : I want 1/8th period
thats a NOGO. formula doesnt allow that. it disrupts . !
Q5 : I want 2/5th period
thats a NOGO. formula doesnt allow that. it disrupts . !
Q6 : I want 2/5th period
I want 7 periods
LIM_end = ( (7 / 1) / 0.5) * 2 = 28 grid units
Area_sin 1 period = ( 2 * Pi * R ) * ( (7 / 1 ) / 0.5 ) * 2 / Pi^2
Area_sin 1 period = ( 6Pi ) *28 / Pi^2 = 53.47 cm^2
double check area of 1 period * 7 = 7.63 cm^2 * 7 periods = 53.47 cm^2
see that 1 wavelenght has half the area of 2 wavelenghts. and that half a wavelenght has half the area as 1 complete. Area of 1/4th wavelenght has 1/4th the area of a complete period.
Concerning the Sinus wave. The wave as it appears on paper is not how it is with sound waves or radio waves, or light waves in real medium. On paper it looks like a snake . But in reality there is a Compression and Rarification of air, if sound waves. A compression and rarification interval of "vacuum", once the lightwave is in space or void of molecules. If Vacuum is the void of molecules and light surely is transmitted, the working component of vacuum I am tempted to say "Aether", commonly used before 1960. However, radiance works in all directions, omnidirectional, unless directed, by a lamp shade, a directed antenna like a Yagi antenna . The sinus is a papertool visualisation, a sinus curve does not emit from a lamp or antenna .
I want to know the Energy in watts per phase, then I need the U_volt * I_ampere, in electrical terms we call that WORK, We measure that in Watt hours or kWh, without the use of caclulus or areas of curves . The concept of Energy is confusing for it might mean HEAT, or it can be considered WORK. Heat in a electric motor is considered a loss, but heat in a steam engine is what makes it work. Another type of energy is in nutriron known as Calories . The Sun, provides heat and allows photosynthesis to the plants, so heat and "electromagnetic radiance" is work by transporting signal of information or heat as IR waves. (in my opinion EM transmission waves should be treated as optics) . Work is usually mechanical or logic work, lifting things up, transport or making things go round, calculate a result or making plants grow . A computer performs logical work .
Anyways, energy or for one phase or for a continous pulstrain over an hour. Also understand the different formulas for combustion and flame energy, as well as electrical energy . A short blast of an electrical capctior, a short blast of an exploding star . Or the continous heat from a torch, and the continious work from an electrical generator . The question boils down to what source of energy is used to put the machine to work. Is it a boiling kettle, under bonfire, heated by nuclear reaction, or heated by coal, steam technology such as Hero`s Engine . A chemical reaction such as a battery, a reciprocal piston engine of gasoline, diesel or BioDiesel combustion, an Otto Engine . Or the newer type of hydrogene stored on tanks from electrolysis, Hydrogene is injected in the cylinder and pistons drives an electrical generator, that feeds the electric motor. Notice about cars after the grand entry of electric cars as of about 2015, that no longer is it mentioned the 0-100 kmph in seconds ( the electric cars accellerate furiously), but now all cars are measured in Kilo Newton. A Stirling motor with hot and cold air exchange into its piston. Or is it an electrical AC generator (solar, hydro or wind) . In other words, what motivates the Prime mover . That would be the elements of Water, Earth, Wind and Fire, and the emptyness between them . The esotheric 5th element "the void".
I realize that the formula is only usable for quarter waves, half waves and whole waves. Or 1/4th, 2/4th , 3/4th and 4/4th of a wave or more. It gives wrong answer to 1/8th waveform, and 1/3rd or 2/5ts . This method will not enable you to calculate the Area under a function, then you must use conventional Calculus.
A_curve (north and south) = Wavelenght * O_circumf_cylinder / Pi^2
A_curve (north and south) = 1cm wavelenght * 31.4 / Pi^2 = 3.18
Example 3
3 SIN (X*pi * 5)
Radius = 3 cm
wavelenght ( lambda ) = 5
Q: Find the area of both north and south curve with respect to baseline.
O_circumf_cylinder = Pi * R * 2 = 3.14 * 3cm* 2 = 18.84
A_Cyl = Lambda * O_Cyl = 5 cm * 18.84 = 94.24 cm^2
A_curve ( north and south) = Lambda * O_cyl / pi^2 = 94.24 / pi^2 = 9.54 cm^2
3 lines of code !
Sum from 0 to 5.
Example 4
2 SIN ( X* pi * 4)
the last number 4, means this. 4 half waves per 1 unit lenght.
to obtain the area of a Phase ( or 1 period ) .
Lenght of half a period = 1 period / 4 = 0.25 cm long northwave
wavlenght 1 phase = (1/4) * 2
Radius = 3 cm
lambda = ( 1/4 ) * 2
A_phase = lambda * O_cyl / pi^2
A_phase = lambda * pi*2R / pi^2
A_phase = lambda * 2R / pi
A_phase = (1/4)*2 * (pi * 2*3 ) / Pi^2 = 0.95 cm^2
A_phase = (1/4)*2 * (pi * 2R ) / Pi^2
A_phase = 0.5 * 6 / Pi = 0.95 cm^2
Final result :
Areal_phase = lambda * 2R / pi
The notation "lambda" works as a "limit", that defines the end of your function. you can put it as long as you want or short as you want. you accumulate the area of the waveperiods. I havent figured out how to move the start limit of the function. Maybe try the SIN (X+10) to phase the waveform sideways .
Example 5
Lets say youn want the area of the waveform for 1 sec duration
3 SIN ( X * Pi * 4)
R = 3
Limit_end = 4 * 1 sec
A_phase_1 sec = Lim * O_cyl / Pi^2
A_Phase_1 sec = 4 * Pi * 2 * R / Pi^2
A_Phase_1 sec = 4 * 2 * R / Pi
A_Phase_1 sec = 4 * 2 * 3 / Pi = 7.63 cm seconds (siffix?)
for 9 seconds
3 Sin ( X * Pi * 4)
R = 3
Lim = 4 * 9 sec = 36
A_phase_1 sec = Lim * O_cyl / Pi^2
A_Phase_1 sec = 36 * 2 * R / Pi
A_Phase_1 sec = 36 * 2 * 3 / Pi = 68.75 cm sec
I got problems when trying to incorperate c= lightspeed 3E8. I realized the X axis either had to be time in seconds or meter. Or else I would want 2 different measures on the X axis, one for seconds and one for meter. X_sec would display 50 periods per second. X_meter would display one waveform at the lenght of c/hz = 3E8 / 50hz = 6 000 000 meter per sec . huh, meter per sec ....
so this didnt work :
U = 230V
lambda wavelenght = c / hz = 3E8 / 50 hz
U Sin (X Pi * (c / hz )) = didnt compute ...
Example 6
more formal terms
f(x) = R SIN (x*pi (N_halfwaves))
f(x) = R SIN (x*pi (hz*2))
Amplitude = Radius
Limit_Lambda_one wavelenght = (1 / hz) * 2 = (1 / 50 hz) * 2 = 0.2 * 2 = 0.04 cm
Hz = 50 periods per second ( 1 sec is 1 grid unit )
Amplitude (Radius) = 5
Area_sinus = Lim 2PiR / Pi^2
Area_sinus = Lim 2R / Pi
Area_sinus_one period = (1/hz)*2 * 2R / Pi
Area_sinus_one period = (1/50hz) * 2 * 2 * 5cm / Pi
Area_sinus_one period = 0.02 * 20 / Pi
Area_sinus_one period = 0.04 / Pi = 0.127 cm^2
Sin_curve lenght = Sinus is converted into a cylinder, then opened into a square. The diagonal on the square is equal to the lenght of the Sinus curve .
Sin_curve lenght = SQRT ( (2PiR)^2 + (1/Hz*2) )
Sin_curve lenght = SQRT ( (2 Pi *5 cm )^2 + (1/50 Hz*2) )
Sin_curve lenght = SQRT ( (31.4)^2 + (0.04)^2 )
Sin_curve lenght = SQRT ( 985 + 0.0016 )
Sin_curve lenght = SQRT 985.0016 = 31.3 cm long
Example 7
f(x) = R sin (x * pi * N_halfwaves)
f(x) = 3 sin (x * pi * 0.5)
R = radius of cylinder or waveform amplitude = 3
N_halfwaves = halfwaves per grid units= 0.5
Lambda = N_halfwaves * 2 = 1 period of 360 degrees rotation .
Lambda can be replaced by Limit or LIM_end .
LIM_end = the intergral version of "to_X". The formula does not apply for "from_X" other than set to 0 .
f(x) = 3 sin (x * pi * 0.5)
Area Sin = 2 Pi R * (LIM_end) / pi^2
Q1 : I want 1 period or a complete wavelenght
1 wavelenght is 2 halfwaves .
LIM_end = 1 period = ( (Desired LIM_end) / N_halfwaves ) * 2 = N grid units
LIM_end = 1 period = ( (1 / 1) / 0.5 ) * 2 = 4 grid units
Area_sin 1 period = (2 * Pi * R ) * ( (1 / 1) / 0.5 ) * 2 / Pi^2
Area_sin 1 period = (6Pi) * 4 / Pi^2= 7.639437268 cm^2
Q2 I want half a period
LIM_end = ( (1 / 2) / 0.5) * 2= 2 grid units
Area_sin 1 period = (2 * Pi * R ) * ( (1 / 2) / 0.5 ) * 2 / Pi^2
Area_sin 1 period = (6Pi) * 2 / Pi^2= 3.81 cm^2
Q3 I want a quarter period
LIM_end = ( (1 / 4) / 0.5) * 2 = 1 grid units
Area_sin 1 period = (2 * Pi * R ) * ( (1 / 4) / 0.5 ) * 2 / Pi^2
Area_sin 1 period = (6Pi) * 1 / Pi^2= 1.90 cm^2
Q4 : I want 1/8th period
thats a NOGO. formula doesnt allow that. it disrupts . !
Q5 : I want 2/5th period
thats a NOGO. formula doesnt allow that. it disrupts . !
Q6 : I want 2/5th period
I want 7 periods
LIM_end = ( (7 / 1) / 0.5) * 2 = 28 grid units
Area_sin 1 period = ( 2 * Pi * R ) * ( (7 / 1 ) / 0.5 ) * 2 / Pi^2
Area_sin 1 period = ( 6Pi ) *28 / Pi^2 = 53.47 cm^2
double check area of 1 period * 7 = 7.63 cm^2 * 7 periods = 53.47 cm^2
see that 1 wavelenght has half the area of 2 wavelenghts. and that half a wavelenght has half the area as 1 complete. Area of 1/4th wavelenght has 1/4th the area of a complete period.
Concerning the Sinus wave. The wave as it appears on paper is not how it is with sound waves or radio waves, or light waves in real medium. On paper it looks like a snake . But in reality there is a Compression and Rarification of air, if sound waves. A compression and rarification interval of "vacuum", once the lightwave is in space or void of molecules. If Vacuum is the void of molecules and light surely is transmitted, the working component of vacuum I am tempted to say "Aether", commonly used before 1960. However, radiance works in all directions, omnidirectional, unless directed, by a lamp shade, a directed antenna like a Yagi antenna . The sinus is a papertool visualisation, a sinus curve does not emit from a lamp or antenna .
I want to know the Energy in watts per phase, then I need the U_volt * I_ampere, in electrical terms we call that WORK, We measure that in Watt hours or kWh, without the use of caclulus or areas of curves . The concept of Energy is confusing for it might mean HEAT, or it can be considered WORK. Heat in a electric motor is considered a loss, but heat in a steam engine is what makes it work. Another type of energy is in nutriron known as Calories . The Sun, provides heat and allows photosynthesis to the plants, so heat and "electromagnetic radiance" is work by transporting signal of information or heat as IR waves. (in my opinion EM transmission waves should be treated as optics) . Work is usually mechanical or logic work, lifting things up, transport or making things go round, calculate a result or making plants grow . A computer performs logical work .
Anyways, energy or for one phase or for a continous pulstrain over an hour. Also understand the different formulas for combustion and flame energy, as well as electrical energy . A short blast of an electrical capctior, a short blast of an exploding star . Or the continous heat from a torch, and the continious work from an electrical generator . The question boils down to what source of energy is used to put the machine to work. Is it a boiling kettle, under bonfire, heated by nuclear reaction, or heated by coal, steam technology such as Hero`s Engine . A chemical reaction such as a battery, a reciprocal piston engine of gasoline, diesel or BioDiesel combustion, an Otto Engine . Or the newer type of hydrogene stored on tanks from electrolysis, Hydrogene is injected in the cylinder and pistons drives an electrical generator, that feeds the electric motor. Notice about cars after the grand entry of electric cars as of about 2015, that no longer is it mentioned the 0-100 kmph in seconds ( the electric cars accellerate furiously), but now all cars are measured in Kilo Newton. A Stirling motor with hot and cold air exchange into its piston. Or is it an electrical AC generator (solar, hydro or wind) . In other words, what motivates the Prime mover . That would be the elements of Water, Earth, Wind and Fire, and the emptyness between them . The esotheric 5th element "the void".
I realize that the formula is only usable for quarter waves, half waves and whole waves. Or 1/4th, 2/4th , 3/4th and 4/4th of a wave or more. It gives wrong answer to 1/8th waveform, and 1/3rd or 2/5ts . This method will not enable you to calculate the Area under a function, then you must use conventional Calculus.
☥ Horus ☥
☥ Ptah ( Pythagoras) ☥
☥ Thoth ☥
☥ Ankh Ra ☥
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