<p> Something else from Smath forum </p>

402e37ef-e394-489f-8d5f-09cdaca0939e 18 4 5 decimal

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Scalar integration

The scalar running variable of integration is 'x'. Notice that if you would integrate over ± x it would double the integration. The 'F' datum is 0 ... in other words: you must range the variable of integration from the datum of F. Observe that Int(z) is asymptotic to the 0X axis at infinity.

u F 11 x 2^+ / :

z Int u F u 0 x int :

Bidirectional Analytics of F(x)

z Int u F u 0 x int ≡ same plot ≡

x F x int

x F x diff

x F

x CumulIntegral x F x int maple :

In "statistics", the convention is to scale the bell shape curve so that its "cumulative integral", i.e: the total area under the bell is = 1, the bell is then called PDF. In this example, s=2/π .

u F u 0 x int maple x 1 x 2^+ /

x CumulIntegral x atan

s 2 π / :

1 ∞ atan / 2 π /

The constant of integration is needed if you want to export CumulInt as a function.

∞ CumulIntegral 1.5708

Bidirectional Analytics of F(u)

z Int x F x F x diff 3 1 sys

Statistical Cumulative Integral

s x atan * s x atan x diff ( * s ∞ atan * s ∞ - atan * 4 1 sys

s ∞ atan * s ∞ - atan * 21 mat 11 - 21 mat

More interesting applied maths

data x F 11 x 2^+ / : U 5 - 5 4.99 - range : i 1 U rows range f i el U i el F : for U f augment 41 line :

Plot range

data

Often, in real projects it is useful to "disambiguate". The cumulative integral is the cumulative area under the function. So, discretize the function: range & mesh as per the project and run the "Finite Differences Integrator". This FD integrator is absolutely accurate if ranged & meshed infinitely ... easy here: we have a function. It is most gorgeous if you just have "data", data you can't approximate with a fake function. And, don't worry, it works if the data are "noisy", even pure random noise. Working with "noisy data", don't commit the mistake of "smoothing" the data if the smoother is not wisely applied.

--- Experiment: in 'U' --- plug a range of your choice

5 - 5 4.99 - range

05 0.01 range

Observe that no matter the range, the cumulative has datum @ '0' ,,, always.

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

Σdif data Σ :

x datum 0 :

Σdif x datum 2 1 sys <p> Something else from Smath forum </p>

u F x 2 - :

z Int u F u 0 x int :

z Int x F 2 1 sys

# # # diff from diff(3) f(x) menu ≡

u F 11 x 2^+ / :

n 0 :

Derivative n=0 is the function

nthDerivative x F x n diff x F x 1 diff x F x 2 diff x F x 3 diff 41 sys :

Successive Derivatives

nthDerivative