If oil is leaking from a pipe at a rate of r(t) = 2t gallons per min at t minutes then you can find the total amount of oil leaked in the first 4 minutes by finding the area under the graph of r(t) from 0 to 4. This is the definite integral of 2t from 0 to 4 which equals t^2 evaluated at 4 minus t^2 evaluated at 0 or 16 minus 0 equals 16.
What meaning if any can be given to the area under the curve of the total oil leaked function R(t) = t^2 from 0 to 4?
Basically what meaning can be given to the area under a function's graph without considering the function as a rate of change function?
Feel free to use different units. For example maybe start with a rate of change function of P'(x) = 5x + 1 representing marginal profit in dollars per unit sold if it helps give meaning to the area under the graph of the antiderivative of the function.