I think there is no solution.

The first (inner) integral is found to be

In[4]:= Integrate[x*e^(-x^2), {x, 0, y*z}] // FullSimplify

Out[4]= -((-1 + e^(-y^2 z^2))/(2 Log[e]))

which is essentially a constant and an exponential Function, a + e[x] say. Insert this in the integrand for the "outer" integral and multiply things out:

In[6]:= (a + e[x])*(y/4*a*b)*(\[Pi] +2 ArcSin[(2*(y^2 - (b - h)^2)/y^2) - 1]) // Expand

Out[6]= 1/4 a^2 b \[Pi] y - 1/2 a^2 b y ArcSin[1 - (2 (-(b - h)^2 + y^2))/y^2] + 1/4 a b \[Pi] y e[x] -  
1/2 a y ArcSin[1 - (2 (-(b - h)^2 + y^2))/y^2] e[x]

The first term is a linear function of y, and the corresponding integral, which is essentially y^2 in the bounds 0 and infinity of course tends to infinity or doesn't converge:

In[7]:= Integrate[1/4 a^2 b \[Pi] y, {y, 0, Infinity}]

During evaluation of In[7]:= Integrate::idiv: Integral of y does not converge on {0,\[Infinity]}. >>

Hi Ratnam,

There are syntax errors in the code e.g.

* [\[Pi] + 2 ArcSin ((2*(y^2 - (b - h)^2)/y^2) - 1)])

Fix the errors, they are highlighted in the notebook in red and in the error messages.

Syntax::sntxf: "(y/4ab)*" cannot be followed by "[[Pi]+2ArcSin((2*(y^2-(b-h)^2)/y^2)-1)]".