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## Homework Statement

Show that the heat energy per unit mass necessary to raise the temperature of a thin slice of thickness [itex]\Delta[/itex]x from [itex]0^o{}[/itex] to [itex]u(x,t)[/itex] is not [itex]c(x)u(x,t)[/itex]. but instead [itex]\int_0^uc(x,\overline{u})d\overline{u}[/itex].

## Homework Equations

According to the text, the relationship between thermal energy and temperature is given by

[itex]e(x,t) = c(x)p(x)u(x,t)[/itex],

which states that the thermal energy per unit volume equals the thermal energy per unit mass per unit degree times the temperature time the mass density.

When the specific heat [itex]c(x)[/itex] is independent of temperature, the heat energy per unit mass is just [itex]c(x)u(x,t)[/itex].

## The Attempt at a Solution

The only hint really is that this is related to the area, from the solution. How can I go about this geometrically and/or algebraically?

Any help/pointers will be much appreciated. Thank you!