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=limn→∞n∑k=1a−bnf(a+ka−bn)=∫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)=ddx∫xaf(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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