Average of Accumulative Physical Quantity

Average Value

Besides the making use a density function to obtain the mean or weighted average of an accumulative physical quantity in a probability distribution, the average value of accumulative physical quantity can also be determined by integration when the physical quantity can be expressed as a continuous function and is quantified in the form of  an infinitesimal element. imply

Then the average value of accumulative physical quantity can then be determined as an average value of a function. Imply

Average temperature of room temperature

For example, a room temperature conditioner is used to regulate a room through a thermostat. The temperature in the room is fluctuating because of the cyclic switching on and off of the conditioner and other external factors. When the room temperature conditioner is set at 25oC, the ambient temperature in the room can be described by a function

Graphically,

By assuming the recorded temperature is infinitesimal elements of a continuous function, the average temperature of the room temperature over a closed interval [0,24] is 

Graphically,

Average Value by integration

The average value can therefore be applied to any accumulative physical quantity which can be expressed as infinitesimal elements of a continuous function by dividing the total physical quantity  obtained by integration on a closed interval over the length of integration. And mathematically, the length of integration can also expressed in the form of integration. Imply

Therefore the average value of accumulative physical quantity is equal to the weighted average of accumulative physical qunatity with equally spaced infinitesimal elements. In other words, the physical quantity is quantified in the form of infinitesimal elements. Imply