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Friday, May 4, 2018

Fundamental of Integral

Fundamental of Integral

I believe when you think about probabillity , understanding in Integral is imperative, since you have to deal with probability density function , merginal probability distribution and list goes and on and on....
Hence, here I'll try to wrap my head around fundamentals of Integral.

1. Antiderivative

In calculus, Antiderivative of function f is differentiable function F whose derivative is equal to the original function f.
F(x) such that ddxF(x)=f(x) is called antiderivative. However always explain this sentence is a little clumsy, thereby there is notation as bellow.
F(x)=f(x)dx
The expression f(x)dx is called indefinite integral.

2. Definite integral

In order to approximate areas under the curve, take the sum of rectangles.
Let us assume that approximate area on the closed interval [a,b], which is S, under the curve defined by f(x), we can write as bellow.
S=limnnk=1abnf(a+kabn)=bbf(x)dx

3. Theorem of calculus

Let us assume function f(t) is continuous on the closed interval [a,b]. Now I labeled horizontal axis as t.

Let x be a point between a and b. The area under the cuve on closed interval [a,x] defined by f(x) would be funcion with respect to x.
Now let F(x) be that function.
F(x)=xaf(x)dt  where x in [a,b]
In this context, Fundamental theorem of calculous says,
ddxF(x)=ddxxaf(x)dt=f(x)
From this theorem,

  • Every continuous f has an antiderivative F(x).
  • Connection between derivatives and Integration.

Next, let us assume the area under the curve on the closed interval [c,d]

The area on the interval [a,c] is caf(t)dt=F(c), on the other side, the area under the curve on the closed interval [a,d] is daf(t)dt=F(d). Therefore,
dcf(t)dt=F(d)F(c)

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