# Simplification of an expression

How do I simplify the following expression? $$\displaystyle \frac{\int_q^1 w(s) \int_0^s e(\xi) d\xi ds}{2\int_q^1 w(s) ds} p$$ where $w(t)$ is nondecreasing $w(t)>0$ on $(q,1]$ , $e :(0,1)\rightarrow \mathbb{R}$ is continuous, $e\in L(0,1)$, and $\displaystyle\frac12<p<q<1$

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i need to simplify this integration $$\displaystyle \frac{\int_q^1 w(s) \int_0^s e(\xi) d\xi ds}{2\int_q^1 w(s) ds} p$$ – Vrouvrou May 16 '13 at 9:49
In which way do you want to simplify it? – Mhenni Benghorbal May 16 '13 at 10:28
i want to transform $\displaystyle \frac{\int_q^1 w(s) \int_0^s e(\xi) d\xi ds}{2\int_q^1 w(s) ds}$ into an integral $ds d\xi$ – Vrouvrou May 16 '13 at 10:35

Changing the order if integration gives $$\int_q^1 w(s) \int_0^s e(\xi) d\xi ds = \int_0^q e(\xi) \int_q^1 w(s)\, ds d\xi + \int_q^1 e(\xi)\int_{\xi}^1 w(s)\, ds d\xi .$$

Note that, the first integral on the right hand side can be written as

$$\int_0^q e(\xi)d\xi \int_q^1 w(s)\, ds = \int_q^1 w(s)\, ds\int_0^q e(\xi)d\xi.$$

Plot the region to see how the first equation derived.

Added: Plot the region $\xi(s) = s$ where $q \leq s \leq 1$. Then to change the order of integration, consider taking a horizontal strip and notice that the horizontal strip will be bounded below by two different functions, namely, $s=q$ and $s=\xi$. Accordingly, find the limits for $\xi$.

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i don't understand , how you have done that ? – Vrouvrou May 17 '13 at 19:44
Beghorbal please – Vrouvrou May 17 '13 at 20:08
ok, we dont use Fubini ? – Vrouvrou May 20 '13 at 14:46