Suppose that we know that 1 is greater than or equal to g(x)dx with bounds 0 to 4 and is equal or greater to 8. Determine the best possible upper and lower bound for x g(x^2)dx with bounds 0 to 2 that can be deduced from this information.
Suppose that we know that 1 is greater than or equal to g(x)dx with bounds 0 to 4 and is equal or greater to 8. Determine the best possible upper and lower bound for x g(x^2)dx with bounds 0 to 2 that can be deduced from this information.
This is just a glorified integration by substitution problem.
Consider $\displaystyle \int_0^2 x g(x^2)~dx$
Let $\displaystyle u = x^2$
$\displaystyle \Rightarrow du = 2x~dx$
$\displaystyle \Rightarrow \frac 12 ~du = x~dx$
Changing the limits we find:
when $\displaystyle x = 2$, $\displaystyle u = 2^2 = 4$
when $\displaystyle x = 0$, $\displaystyle u = 0^2 = 0$
So our integral becomes:
$\displaystyle \frac 12 \int_0^4 g(u)~du$
I leave the easy part to you
EDIT: Well thanks for making me type all this, Moo