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.

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- May 4th 2008, 10:56 PMHibijibiUpper and lower bounds of integration.
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.

- May 4th 2008, 11:22 PMMoo
- May 4th 2008, 11:23 PMJhevon
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