As we know from the Fundamental Theorem of Calculus, specifically part two, the area between the function f(t) and the x-axis (which also can be referred to as "the area under the curve") can be found by finding the change in the antiderivative of f(t) on the interval [a,b]. As the interval grows larger, the area under the curve can be said to accumulate. In the simpler words of Riemann sums, infinitely thin rectangles add up to approximate the area between a function and the x-axis, so logic dictates that as the interval [a,b] grows larger the tiny rectangles that represent area accumulate.

The following image illustrates this point:

In this case, negative fragments of the area accumulate until the function crosses the x-axis, after which point the positive area fragments continue to accumulate. The addition of positive and negative values relative to the x-axis is clearly in compliance with the established techniques for evaluating definite integrals.

Example and Practice Problems:

Basic steps for solving Accumulation problems:

1.Find the rate function, usually by finding the derivative of a given function. For a large portion of Accumulation problems, this step is unnecessary because the rate function is already given. Remember to make sure that the units of the rate function make sense (i.e. people per hour).
2. Set up a definite integral of the rate function, typically with respect to time (t). Some additional work may be required to determine the proper upper and lower bounds (b and a, respectively).
3. Evaluate the integral to determine the area between the rate function and the x-axis. Typically, the given rate function will be extremely difficult to evaluate with out the use of a calculator. So, use a calculator to evaluate the integral by entering the function, variable, and both bounds.
***NOTE: As in one of the following example problems, if the original function is given, the accumulated variable can be found simply by finding the difference between f(b) and f(a).

Factors that could complicate the basic accumulation problem:

1. Problems, such as those involving theme park populations, that involve more than one rate function are slightly more complicated than one-rate scenarios. Often the two rate functions will conflict, i.e. one will increase and the other will decrease, or one will be positive and the other will be negative. These situations will possibly require the student to find the difference between the areas under the curves.
2. Some problems may involve the use of graph theory, relating first and second derivatives with the original function.

Additional Resources: (PDF file)