So I'm a calculus student who isn't that far along yet (about to end Calc II at uni) and I was looking at integrals in my spare time...
Basically my question is this: $$ \int_0^af(x)dx =0.5 $$
Where a = all real numbers except 0 and I'm trying to find an answer to f(x)
The reason I thought of this is because $\int_0^1xdx=0.5$ and I was wondering if there was some sort of circle or circle-like thing (turns out there's no circle to describe this) or curve upon which all input x values resolve to have an integral of 0.5
This also might get hard as $\lim_{x\to0}f(x)$ should start to act weird, since the distance between x and 0 are super small... I wonder if this is even possible.
Also if you have time, I'd wanna find out if there's a generalizable method I can use to always find f(x) given any value (other than 0), not just 0.5
I also will specifically request that whatever answer I get be dumbed down to a college sophomore level. Thank you!!!
$\int_0^x f(t)\ dt = F(x)$ where $\frac {d}{dx} F(x) = f(x).$ This is the fundamental theorem of calculus.
If $F(x)$ is constant for all $x > 0$ then $f(x) = 0$ for all $x>0$