Entropy change at varying temperatures?

Entropy change is defined as the amount of energy dispersed reversibly to or from the system at a specific temperature. Reversivility means that the temperature of the system must remain constant during the dispersal of energy to or from the system . But this criterion is only fulfilled during phase change & isothermal processes. But not all processes maintain constant temperature;temperature may change constantly during the dispersal of energy to or from the system . To measure entropy change ,say, from $300K$ & $310K$, the range is divided into infinitesimal ranges ,then entropy is measured in that ranges and then is integrated as said in this site . But I cannot understand how they have measured entropy change in that infinitesimal ranges as there will always be difference between the temperature however small the range might be . What is the intuition behind it? Change of entropy is measured at constant temperature,so how can it be measured in a range ? I know it is done by definite integration but can't getting the proper intuition . Also ,if by using definite integration to measure change, continuous graph must be there(like to measure change in velocity,area under the graph of acceleration is measured) . So what is the graph whose area gives change in entropy? Plz help me explaining these two questions.

For a reversible addition of heat, the entropy change is $\int \frac{dQ} {T}$, in other words the area under the graph of $\frac{1}{T}$ against $Q$ (heat added to system).
And yes, when a small amount of heat $\Delta Q$ is added, temperature T changes only a little, so $\frac{\Delta Q} {T}$ is well-defined. When added up this gives the integral.
Calorimetry is a nasty business. Whenever is possible entropy is measured by measuring mechanical parameters by taking advantage the Gibbs form of 2nd law. For example, if a thin rod is elastic then $dU = TdS+\sigma d\epsilon$ where $\sigma, \epsilon$ are the stress and strain. From the equality of derivatives you get $\frac {\partial S}{\partial \sigma}|_T = -\frac {\partial \epsilon}{\partial T}|_\sigma$ the right side of which you can measure directly and then you can integrate with respect to stress at constant $T$. Thus you only have to measure entropy as function of stress at one temperature, the rest you can get by measuring thermal expansion at constant stress.