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 $\frac{d}{dx} F(x) = f(x)$ is called antiderivative. However always explain this sentence is a little clumsy, thereby there is notation as bellow.
$$F(x) = \int f(x) dx$$
The expression $\int f(x)dx$ 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.
$$S = \lim_{n \to \infty} \sum_{k=1}^{n} \frac{a-b}{n}f(a+k\frac{a-b}{n}) = \int^{b}_{b}f(x)dx$$

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.
$$F(x) = \int^{x}_{a} f(x) dt \ \ where\ x\ in\ [a,b]$$
In this context, Fundamental theorem of calculous says,
$$\frac{d}{dx} F(x) = \frac{d}{dx} \int^{x}_{a}f(x) dt = f(x)$$
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 $\int^{c}_{a}f(t)dt = F(c)$, on the other side, the area under the curve on the closed interval [a,d] is $\int_{a}^{d} f(t) dt = F(d)$. Therefore,
$$\int^{d}_{c}f(t) dt = F(d) - F(c)$$