Asking about integral notation

[songoku]

TL;DR

Why writing $$\int_{a}^{x} f(x) dx$$ is not correct?

Why should it be $\int_{a}^{x} f(t)dt$ ?

Couldn't it be like this:
Let F(x) = $\int f(x)dx$ so $\int_{a}^{x} f(x)dx$ = F(x) - F(a)

Thanks

 

[fresh_42]

Summary: Why writing $$\int_{a}^{x} f(x) dx$$ is not correct?

Why should it be $\int_{a}^{x} f(t)dt$ ?

Couldn't it be like this:
Let F(x) = $\int f(x)dx$ so $\int_{a}^{x} f(x)dx$ = F(x) - F(a)

Thanks

The integral $\displaystyle{\int_a^b f(t)\,dt}$ is short for $\displaystyle{\int_{t=a}^{t=b} f(t)\,dt}.$ If you use the same letter ($b=t$) for two different meanings then you cause confusion.

 

[PeroK]

Summary: Why writing $$\int_{a}^{x} f(x) dx$$ is not correct?

Why should it be $\int_{a}^{x} f(t)dt$ ?

Couldn't it be like this:
Let F(x) = $\int f(x)dx$ so $\int_{a}^{x} f(x)dx$ = F(x) - F(a)

Thanks

Why is it not correct to write $$\sum_{k = 1}^k a_k$$

 

[songoku]

Thank you very much for the explanation fresh_42 and PeroK