CPM Homework Help : CALC Problem 6-127

Suppose that $C(t)$ represents the cost per day to heat your house measured in dollars per day, where $t$ is time measured in days and $t = 0$ corresponds to January 1, 2000. Interpret the expressions below:

$\int _ { 0 } ^ { 31 } C ( t ) d t$ and $\frac { 1 } { 31 } \int _ { 0 } ^ { 31 } C ( t ) d t$

Hint:

The second expression divides the first integral by $31$ . What does $31$ represent? Why divide?
What practical information will it calculate about the cost of heat?

Answer:

The first integral represents the area under the $C(t)$ curve between $t = 0$ and $t = 31$ .
Since $C(t)$ represents cost per day, its a rate function.
The area under this rate function represents the accumulated cost during the $31$ days of January.