Issues with an application of definite internal and marginal utilities
The mistake you are making appears to be treating $\theta_i$ as a variable while in reality it is a specific value of the derivative.

So 
$$\int_0^{q_i} V'(q)dq = \int_0^{q_i} \theta_i dq \;\;\; i=l,h$$

is wrong, since $V'(q)$ equals $\theta_l$ evaluated at $q_l$ only (and equals $\theta_h$ evaluated at $q_h$ only).