Newton Leibnitz Formula for Evaluating Definite Integrals

[andyrk]

Lately, I have been trying really hard to understand the Newton Leibnitz Formula for evaluating Definite Integrals. It states that-
If f(x) is continuous in [a,b] then $\int_a^b f(x) dx = F(b) - F(a)$.
But one thing that just doesn't make sense to me is that why should f(x) be continuous in [a,b] if we need to apply this formula?
Reply soon!

 

[andyrk]

Anybody there?

 

[AURUM]

do you know limit as as sum formula?

 

[HallsofIvy]

It doesn't say that. The statement "if a then b" means "if a is true then b is true". It does NOT say anything about what happens if the hypothesis is NOT true.

This theorem says that "if f is continuous on the interval [a, b], then $\int_a^b f(t)dt= F(b)- F(a)$". It does NOT say anything about what happens if f is NOT continuous, If f is not continuous, then this may or may not be true.

 

[andyrk]

Its related to the Fundamental Theorem of Calculus which states - $\int_a^x f(x) dx = F(x)$.
The explanation is given here: https://www.khanacademy.org/math/in...t-and-second-fundamental-theorems-of-calculus

 

[AURUM]

please read this-http://www3.ul.ie/~mlc/support/Loughborough%20website/chap15/15_1.pdf

 

[AURUM]

read it fully and pay attention to the formula of the area.