# Integral not simplifying

I specify a function in terms of an integral and then try to evaluate it with Simplify. However, the answer is not really simplified to what it should be.

assumptions := {l > 0, Element[NN, Integers], NN > 0, b > 0, a > 0,
l > b, a > l, α > 0, β > 0};

Φ = Function[x,
If[x < l,
(NN*π*α*b^2 *(l^2 ArcCos[x/l] - x Sqrt[l^2 - x^2]))/(a^2 * β), 0]
];
Px = Function[x, If[0 <= x <= a, 1/a, 0]];
FΦ = Function[φ, Simplify[Integrate[If[Φ[x] <= φ, 1, 0] Px[x], {x, 0, a}],
assumptions]];
Simplify[FΦ[0], assumptions]


However, it is easy to see that it should be reduced as first

$\frac{1}{a} \int_0^a \text{If}\left[\text{If}\left[x<l,\frac{\text{NN} \pi \alpha b^2 \left(l^2 \text{ArcCos}\left[\frac{x}{l}\right]-x \sqrt{l^2-x^2}\right)}{a^2 \beta },0\right]\leq 0,1,0\right] \, dx$

and then $\frac{1}{a} \int_0^l \text{If}\left[\frac{\text{NN} \pi \alpha b^2 \left(l^2 \text{ArcCos}\left[\frac{x}{l}\right]-x \sqrt{l^2-x^2}\right)}{a^2 \beta }\leq 0,1,0\right] \, dx + \frac{1}{a} \int_l^a \text{If}\left[ 0 \leq 0,1,0\right] \, dx$

to $\frac{1}{a} \int_0^l 0 \, dx + \frac{1}{a} \int_l^a 1 \, dx$ $=\frac{a-l}{a}$.

How can I get Mathematica to simplify it correctly?

• I am not sure if this helps and if I am right. But I think you are wrong in your second line with the first term. The Integration is done after the If. Without any value for x you (and Mathematica) cannot decide if this term is ≤ 0 or not. E.g. if you use NN = 1; b = 2; α = 1; β = 1; l = 5; a = 6; (this should meet your assumptions), then the term ist only 0 for x=5. Mar 20, 2012 at 22:04
• P.S.: If I am right, then you are also wrong in your first line because of the same mistake (you do not have a x, so you do not know if x<l or not). Mar 20, 2012 at 22:10
• @partial81, In assumptions, we know that $a>l>0$, so the integral from $0$ to $a$ can be broken into $0$ to $l$ and $l$ to $a$. Then, $x<l$ for the first part and $x>l$ for the second part. The values you give don't satisfy $a>l$. Mar 20, 2012 at 22:24
• $a=6$; $l=5$; $=>$ $a>l$. What is wrong with my example values? Mar 21, 2012 at 17:23

Mathematica obviously cannot decide the sign of l^2*ArcCos[x/l] - x*Sqrt[l^2 - x^2]. The rest of your code is fine. For example, just replace the first function by
Function[x,

(in other words, just get rid of -x*Sqrt[l^2 - x^2]) and everything will work as expected.