Question

Let \({\rm{r}}\left( {\rm{t}} \right)\) be the rate at which the world's oil is consumed, where t is measured in years starting at \({\rm{t = 0}}\) on January 1 , 2000 , and \({\rm{r}}\left( {\rm{t}} \right)\) is measured in barrels per year. What does \(\int_{\rm{0}}^{\rm{3}} {\rm{r}} {\rm{(t)dt}}\) represent?

Short answer

The net change in global oil consumption in barrels from year 0 to year 3 is \(\int_0^3 r (t)dt\).

Step-by-step solution

Definition the Net Change Theorem

The integral of a rate of change is considered in the net change theorem. It states that when a quantity changes, the new value is equal to the initial value plus the integral of that quantity's rate of change. There are two methods to express the formula. The second is the definite integral, which is more familiar.

Finding the Net change

According to Net change theorem,

If\({\bf{r}}\left( {\bf{t}} \right)\)is the rate of consumption of world's oil,\(R\left( t \right)\), with respect to year,\(t\).

Then, \(\int_0^3 r (t)dt = R(3) - R(0)\)

As a result, the net change in global oil consumption in barrels from year 0 to year 3 is \(\int_0^3 r (t)dt\).