Looking for interesting examples of when this integral identity is satisfied.

Question

empty · Answer

$$\left( \int_a^b f(x) dx \right)^2 = \int_a^b [ f(x)]^2 dx$$

empty · Answer

I have found two sorta separate, but specific cases of stuff. Just playing around lol.

dan815 · Answer

im just pondering my hands are too messy to do anything lol i got cheetos and doritors and ruffles

empty · Answer

Haha well I'll give you a more general question to ponder, what are all the pairs of functions that obey this relationship:

$$g\left( \int_a^b f(x) dx \right) = \int_a^b g[ f(x)] dx$$

Clearly $g(x)=Cx$ and $f(x) = \text{any function}$ is one such pair. But what about other functions $g(x)$? I want to be able to find the pairing $f(x)$ given $g(x)$. Suck on that, 3-chips-kid. ;P

empty · Answer

I suppose the dependence on the bounds of the integral might also be something to consider as well.

dan815 · Answer

lets just considering some geometric shapes for now

empty · Answer

Sounds good, I tried to do that but sorta failed at visualizing $\int f^2(x)dx$

dan815 · Answer

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