Prove that if functions f and g are continuous on a closed interval [a, b] and if f(x) ≥g(x) on

Prove that if functions f and g are continuous on a closed...

Key Concepts

Monotonicity of the Integral

The monotonicity property of the integral states that if one function is greater than or equal to another function at every point in an interval, then the integral of the first function will be greater than or equal to the integral of the second over that interval. This concept is crucial because it allows one to compare the cumulative values of functions directly based on pointwise comparisons, leading to proofs like the one in the question.

Continuity on Closed Intervals

A function that is continuous on a closed interval [a, b] is guaranteed to be Riemann integrable due to the extreme value theorem and other results in real analysis. This concept is essential because continuity ensures that the function does not have any discontinuities or erratic behavior that could disrupt the accumulation process integral to the notion of integration.

Riemann Integration

Riemann integration is a method of assigning a numerical value, called an integral, to a function over an interval by approximating the area under its curve using sums of areas of rectangles. The rigorous definition involves partitioning the interval, forming upper and lower sums, and taking the limit as the mesh of the partition tends to zero. This process lays the foundation for the mathematical analysis of accumulation and area, and it is applicable to continuous functions on closed intervals.

Transcript

00:01 In this question, it is given that f and g are two continuous functions on the close interval a to b.

00:08 It is given if f of x is always greater than equal to g of x on the interval a to b.

00:15 We need to prove that the integral of the function f of x for the limit a to b will also be greater than equal to the integral of function g of x for the limit a to b.

00:25 Now, from the definition of definite integrals, we have, that is the definite integral of f of x, that is integral f of x, dot dx for the limit a to b is equal to, is actually equal to, that is the sum of, that is limit of sum, limit, n tends to infinite, a limit on the, that is, that is, f of u .i times delta x and i is from 1 to n now we know that f of x is greater than equal to 0 and delta x will also be greater than 0 now from these two values we will get that f of ui times delta x will also be greater than 0 that is therefore summation of f of ui times delta x will also be greater than 0 for equal to 1 to n now from here by the properties of the limits we will have that is limit n tends to infinite on the that is summation of f of ui times delta x for the limit that is for i equal to 1 to n will be greater than or equal to limit on that is limit of n tends to infinite on zero which is also equal to zero therefore from here we can conclude that this limit of sum which is that is limit and tends to infinite summation of f of ui times delta x for i equal to 1 to n will be greater than equal to 0 now we can see on the left -hand side this expression is just equal to the, that is definite integral of function f of x for the limit a to b.

02:41 Therefore, we will get from here that that is the integral of f of x for the limit a to b.

02:50 Dx will also be greater than equal to 0.

02:53 So from here, we got that if the function is greater than equal to 0, then its integral will also be greater than equal to 0...

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