The above example assumes that the expansion coefficient did not change as the temperature changed and the increase in volume is small compared to the original volume. This is not always true, but for small changes in temperature, it is a good approximation. If the volumetric expansion coefficient does change appreciably with temperature, or the increase in volume is significant, then the above equation will have to be integrated: ln ⁡ ⁡ ( V + Δ Δ V V ) = ∫ ∫ T i T f α α V ( T ) d T Δ Δ V V = exp ⁡ ⁡ ( ∫ ∫ T i T f α α V ( T ) d T ) − − 1 where α α V ( T ) is the volumetric expansion coefficient as a function of temperature T, and T i and T f are the initial and final temperatures respectively.