It is known that solids expand on heating. Consider a case of rod of negligible area of cross-section such that it expands linearly upon heating.Now the increment in the length of the rod is proportional to the change in temperature as well as its length... Thus: $$dl=l*\alpha*\Delta T———(a)$$ ...where $\alpha$ = coefficient of linear expansion of the material with which the rod is made with.
Consider a rod (of length $l_o$)with a variable $\alpha$ such that $\alpha = Kx$...where x is the distance of a point on the rod from one of its end.calculate the the final length of rod when the temperature of the rod is increased by $\Delta$T Now using (a) $$dl=dx*kx*\Delta T———(b)$$ next step is to integrate (b).. $$\int dl =\int kxdx*\Delta T$$
My doubt is regarding the limits to be taken for x.Is it from o to $l_o$?If so what about extra length which is generated slowly by the change in temperature...shouldn’t we include the expansion of that part in to consideration?