Integral not simplifying

Question

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?

Answer 1

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, 
 If[x < l, (NN*Pi*α*b^2*(l^2 ArcCos[x/l]))/(a^2*β), 0]] 

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