A student studies the temperature distribution along a thin metal rod of length 0.8 m. Let T(x) denote the temperature (in kelvins) at a point x meters from the left end. The student has experimentally determined that the derivative T′(x) is continuous and that T′(x) is measured in kelvins per meter.
In a report, the student writes:
“Integrating T′(x) from x=0 to x=0.8 produces a quantity with units of kelvins per meter times meters, which simplifies to kelvins. Because this quantity has the same units as temperature, it is valid to add the result of the integral directly to T(0.4) to obtain the temperature at x=0.8.”
(a) Identify the conceptual error in the student’s reasoning. Your answer should clearly reference the units involved and the meaning of the integral.
(b) Provide a correct description of what ∫00.8T′(x)dx represents for this rod.
(c) Write an appropriate numerical expression—involving T(0), T(0.4), and an integral of T′—for computing T(0.8). Explain why this expression is dimensionally valid.
(d) Suppose T(0)=300 K and ∫00.8T′(x)dx=−15 K. Determine T(0.8) and interpret the sign of the integral in the physical context.