This is possibly a better question for customer care but I want first to know whether the bug was fixed in higher versions of Mathematica. The following code was tested in Mathematica 10.0.2 and 9.0.1. Maybe someone can test it in a higher (or just different) version. As one can see, it does not work. And it does not give an error message but an erroneous result.

If in the integration limits gives erroneous results

In[1]:= f[x_, y_] = a + b x + c y;

Definition of integration limits; it checks out

In[2]:= g[x_] = If[x < r/2 \[Or] x > s - r/2, r/2, (s - r)/2];
Simplify[g[(s - r)/2], Assumptions -> r > 0 \[And] r > s/2 \[And] r < s]
Simplify[g[s/2], Assumptions -> r > 0 \[And] r > s/2 \[And] r < s]
Simplify[g[(s + r)/2], Assumptions -> r > 0 \[And] r > s/2 \[And] r < s]

Out[3]= r/2

Out[4]= 1/2 (-r + s)

Out[5]= r/2

In[6]:= h[x_] = If[x < r/2 \[Or] x > s - r/2, s - r/2, (s + r)/2];
Simplify[h[(s - r)/2], Assumptions -> r > 0 \[And] r > s/2 \[And] r < s]
Simplify[h[s/2], Assumptions -> r > 0 \[And] r > s/2 \[And] r < s]
Simplify[h[(s + r)/2], Assumptions -> r > 0 \[And] r > s/2 \[And] r < s]

Out[7]= -(r/2) + s

Out[8]= (r + s)/2

Out[9]= -(r/2) + s

Symbolic definite integral; it gives the wrong result; assumptions do not make any difference

In[10]:= Integrate[f[x, y], {x, (s - r)/2, (s + r)/2}, {y, g[x], h[x]}, 
 Assumptions -> r > 0 \[And] r > s/2 \[And] r < s]

Out[10]= -(1/2) r (r - s) (2 a + (b + c) s)

In[11]:= Integrate[f[x, y], {x, (s - r)/2, (s + r)/2}, {y, g[x], h[x]}]

Out[11]= -(1/2) r (r - s) (2 a + (b + c) s)

Separate symbolic definite integrals; the correct result

In[12]:= I1 = Integrate[f[x, y], {x, (s - r)/2, r/2}, {y, r/2, s - r/2}];
I2 = Integrate[f[x, y], {x, r/2, s - r/2}, {y, (s - r)/2, (s + r)/2}];
I3 = Integrate[f[x, y], {x, s - r/2, (s + r)/2}, {y, r/2, s - r/2}];
Simplify[I1 + I2 + I3]

Out[15]= -(1/2) (2 a + (b + c) s) (3 r^2 - 4 r s + s^2)