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: where is the volumetric expansion coefficient as a function of temperature T, and and are the initial and final temperatures respectively. Isotropic materials For isotropic materials the volumetric thermal expansion coefficient is three times the linear coefficient: This ratio arises because volume is composed of three mutually orthogonal directions. Thus, in an isotropic material, for small differential changes, one-third of the volumetric expansion is in a single axis. As an example, take a cube of steel that has sides of length . The original volume will be and the new volume, after a temperature increase, will be We can easily ignore the terms as ΔL is a small quantity which on squaring gets much smaller and on cubing gets smaller still. So The above approximation holds for small temperature and dimensional changes (that is, when and are small), but it does not hold if we are trying to go back and forth between volumetric and linear coefficients using larger values of . In this case, the third term (and sometimes even the fourth term) in the expression above must be taken into account. Similarly, the area thermal expansion coefficient is two times the linear coefficient: This ratio can be found in a way similar to that in the linear example above, noting that the area of a face on the cube is just . Also, the same considerations must be made when dealing with large values of . Put more simply, if the length of a cubic solid expands from 1.00 m to 1.01 m, then the area of one of its sides expands from 1.00 m2 to 1.02 m2 and its volume expands from 1.00 m3 to 1.03 m3. Anisotropic materials Materials with anisotropic structures, such as crystals (with less than cubic symmetry, for example martensitic phases) and many composites, will generally have different linear expansion coefficients in different directions. As a result, the total volumetric expansion is distributed unequally among the three axes. If the crystal symmetry is monoclinic or triclinic, even the angles between these axes are subject to thermal