What is the integral of the constant function

avissidep
2021-06-01
Answered

What is the integral of the constant function

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Alix Ortiz

Answered 2021-06-02
Author has **109** answers

Step 1

Integration of a function gives the result as the area covered by the function with an appropriate axis. INtegrals are of two types, definite and indefinite integrals.

Definite integrals are defined for a limit of variable from a lower limit to an upper limit. This is represented by the expression:-

Here, F(x) is the anti-derivative of the function f(x).

Step 2

The value of the definite integral is constructed by using the limits of x from -2 to 2, and the limts of y from 3 to 4 as

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I have a continious function f that is strictly increasing. And a continious function g that is strictly decreasing. How to I rigorously prove that $f(x)=g(x)$ has a unique solution?

Intuitively, I understand that if I take limits to infinity, then f grows really large and g grows very small so the difference is less than zero. If I take the limits to negative infinity then the opposite happens. Using the intermediate value theorem, there must be an intersection, and since they are strictly increasing/decreasing, only one intersection happens.

My question is how do I apply the intermediate value theorem here? I don't have the interval to apply it on. I don't know when f crosses g and therefore can't take any interval. Or is there some sort of axiom applied here that I am missing?

I have a continious function f that is strictly increasing. And a continious function g that is strictly decreasing. How to I rigorously prove that $f(x)=g(x)$ has a unique solution?

Intuitively, I understand that if I take limits to infinity, then f grows really large and g grows very small so the difference is less than zero. If I take the limits to negative infinity then the opposite happens. Using the intermediate value theorem, there must be an intersection, and since they are strictly increasing/decreasing, only one intersection happens.

My question is how do I apply the intermediate value theorem here? I don't have the interval to apply it on. I don't know when f crosses g and therefore can't take any interval. Or is there some sort of axiom applied here that I am missing?

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