stochastic calculus - Girsanov's Theorem - Change of Measure

I have trouble understanding Girsanov's theorem. The Radon Nikodym process $Z$ is defined by:

$$Z(t)=\exp\left(-\int_0^t\phi(u) \, \mathrm dW(u) - \int_0^t\frac{\phi^2(u)}{2} \, \mathrm du\right)$$

Now $\hat P$ is a new probability measure. The trouble is I am not understanding how to go from old $P$ to the new one. The old $P$ is normally distributed with mean $0$ and variance $t$. Now say I want to know the new probability for an infinitesimally small interval around $0.2$. For that I need to know the value of $Z$ at this interval (event you may say). And then I can multiply (integrate) the value of $Z$ with old $P$, and get new $\hat P$.

Assume $t$ is fixed.

I have no idea how to calculate the value of $Z$ for this interval/event. Help would be appreciated.