homework and exercises - Distribution of normal force on a book resting on the edge of a table

Assume a uniform book is resting on a table. The book has length $1$. But not the entire book is resting on the table, a bit $\alpha$ is sticking out of the table.

Obviously, due to experience, we get that if $0\leq\alpha\leq 1/2$, then the book will just remain at rest. If $\alpha>1/2$, then the book will experience a nonzero torque and eventually fall.

But how do we obtain this same answer with physics? Here's a first attempt that apparently does not work out. Assume for simplicity that $\alpha=1/4$, then $3/4$ is resting on the table.

The book is experiencing a normal force and a weight force. To find the weight force, it is sufficient to take a weight force at the center of the book of magnitude $mg$. The contribution to the torque is then $mg(1/4)$ (since the center of mass is $(1/4)$ away from the rim of the table). But how to find the torque coming from the normal force? by Newton's law, the magnitude of the normal force equals the magnitude of the weight, namely $mg$. Now I try to split up the book into infinitesimal components and integrate. I obtain $$\int_0^{3/4} mgxdx = \frac{9mg}{16}$$ This does not at all equal the contribution of the torque, $mg/4$. So there will be a nonzero torque, which can't happen.

So what is going wrong? Is the normal force somehow not uniform over the surface of the table? If it is nonuniform, how is it distributed, and how do I solve this problem as above using integration?

By the way, just for information, I do know how to solve this problem the following way: if the book is somehow a lever resting on just the rim (and not the table), then we can solve this problem ver easily using the law of the lever and such. When we do this, we find eventually that if $0\leq\alpha\leq 1/2$, then the torque must be "negative". The constraint force of the table prohobits this and just sets the torque at $0$. I'm sorry if this doesn't make much sense, but it isn't my main question anyway. My question is why the naive integration answer does not work and if the normal force is somehow not uniform.