Fundamental Theorems of Integration

[Toramana]

We know there are three ways to find integrals (i) geometric (2) analytic (3) applying anti-differentiation that derives from the Fundamental Theorems of Integration (there are two theorems-Newton, Leibniz developed independently : )? Do you know, what is the basic concept of these theorems?

[RevolutionThroughReason]

This is also called Fundamental Theorem of Calculus ...it proves that for continuous functions, the anti derivative exists.

[Hakim Khan Urgonai]

I thought this was about the integration of people.

Toramana, being the integrated Canadian he is, could perhaps shed some light on that too, for us ignorant Wahhabi Muslims.

[sahil112]

We know there are three ways to find integrals (i) geometric (2) analytic (3) applying anti-differentiation that derives from the Fundamental Theorems of Integration (there are two theorems-Newton, Leibniz developed independently : )? Do you know, what is the basic concept of these theorems?

newtons method is used when there is no straightforward method of solution so the next best alternative is a numerical solution.

example the equation :

x^4 + x^3 -1=0

newtons method teaches us to first find a value say x=x(0) which is close enough the to the solution ( you can do this by drawing a rough graph) and then you basically carry out a step by step kind of process until the desired accuracy is obtained using the formula:

x(n+1)=x(n) - f(x(n))/f'(x(n)) for n=0,1,2,3....

[sahil112]

the other i will touch on inabit just got to eat aha

[Toramana]

This is also called Fundamental Theorem of Calculus ...it proves that for continuous functions, the anti derivative exists.

Good...More specifically, proves that the (first) derivative of a function representing area under a curve (i.e., Area function e.g. A(x)) is the function (e.g. f(x) ) bounding the area ---and that, the anti-derivative of f(x) then is the integral over a closed interval (e.g., [a,b]) giving the area function. Remarkable discovery by Newton and Leibniz.

[Toramana]

newtons method is used when there is no straightforward method of solution so the next best alternative is a numerical solution.

example the equation :

x^4 + x^3 -1=0

newtons method teaches us to first find a value say x=x(0) which is close enough the to the solution ( you can do this by drawing a rough graph) and then you basically carry out a step by step kind of process until the desired accuracy is obtained using the formula:

x(n+1)=x(n) - f(x(n))/f'(x(n)) for n=0,1,2,3....

Newton's is a numerical method I guess not very efficient and general (there are other methods e.g., Simpson's rule, Trapzoid rule, etc.). The anti-derivative method is the most efficient.

[RevolutionThroughReason]

actually, I would give credit for the development of Integration to Ibn Haytham.

[RevolutionThroughReason]

Toramana, it seems you're concentrating on a great deal of early undergraduate level mathematics, which is fine. Any reason for that though?

[Toramana]

Toramana, it seems you're concentrating on a great deal of early undergraduate level mathematics, which is fine. Any reason for that though?

Actually, I learn or re-learn mathematics on the fly as I need it for my purpose---Have to recall some early calculus learning to grasp some relatively advanced topics more fully---am doing it for fun as well.

I was good at maths as for as "formula mathematics" as they teach it back home is concerned but didn't know the theory at its basis very well---Can't go even a single step ahead until grasping the basic concept.

[RevolutionThroughReason]

that's very true, fair enough. Need any help let me know, I used to be a tutor.

[Toramana]

that's very true, fair enough. Need any help let me know, I used to be a tutor.

Thanks---sure if needed help.