True or false? If f and g are continuous and f(x) is greater than or equal to g(x) for a≤x≤b then ∫ from a to b f(x)dx ≥ ∫ from a to b g(x) dx. explain your reasoning as to why it's true or false and maybe give an example.

Question

True or false? If f and g are continuous and f(x) is greater than or equal to g(x) for a≤x≤b then ∫ from a to b f(x)dx ≥ ∫ from a to b g(x) dx.  explain your reasoning as to why it's true or false and maybe give an example.

datanewb · Answer

Well, let's start with the example first. Let's take say:

$$g(x) = 3\f(x) = 7$$
Then $$\int_a^bg(x)dx = 3b-3a$$
and $$\int_a^bf(x)dx = 7b-7a$$, so as long as b > a $$\int_a^bf(x)dx = 7b-7a >3b-3a = \int_a^bg(x)dx$$  Also, if you define the integral as the area under the curve from a to b, then it is very clear. (See drawing)

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