# How to calculate Average temperature from area under the curve?

Let us say I have a graph of temperature against time for a material as shown here.

I want to calculate the average temperature. Will it be simple mean of the curve or rms of the curve?

Edit: The material is some metal.

• Will it be different in 'control' or 'measure' cases? Which is correct for control and which is correct for measure? – scico111 Sep 16 '20 at 9:45
• if you are dealing with a polymer then you can't simply integrate, you should integrate temperature multiplied by a shift factor, you would obtain an equivalent time from which you can then compute the average temperature the material "lives" during the cycle – Alberto Brambilla Sep 16 '20 at 10:38
• The material is made up of some metal. – scico111 Sep 16 '20 at 13:02

Let's call the horizontal axis m and the vertical axis T.

Consider a very small slice of the temperature curve starting from the time m to time m+dm. The area of this slice is $$A= dm*T$$, and its center of geometry is at the $$T_{cg}=T/2$$.

Then the average temperature of the 1 cycle is, just using the CG of the different segments and use the following eq i:

$$T_{average} =\frac {\Sigma(A_n *L_{cgn})}{A_{total}}$$

You already mentioned in the question that you want is average, which is mean and not RMS value, use RMS values if the negative sign needs to be treated as positive temperature only.

Here you simply need to calculate the area under the curve (with sign), cummulate it and divide it by the total time period.

Will it be simple mean of the curve or rms of the curve?

The average temperature will involve a simple mean of the curve multiplied by the total time span of the curve.

If you have sampled data at a constant rate (i.e. the time $$dt$$ between data points is constant), then the average of the temperature will simply be the average of all the temperatures.