Accumulation and net change
An instrument reports only how fast an amount is changing. How do you get back the amount itself, its total change, and its largest value?
Where you are 3 inside
What this area is about
Many quantities are measured as rates: cars per hour, kilowatts, velocity. The integral of a rate over a time interval is the net change in the amount, and adding that change to a starting value gives the amount at any moment. This area puts the fundamental theorem of calculus to work on quantities in context.
How these fit together
All three apply the fundamental theorem of calculus: if F' is continuous on [a, b], then \int_a^b F'(t)\,dt = F(b) - F(a), and for continuous f the function \int_a^x f(t)\,dt has derivative f(x).
Amounts from rates. The net change theorem in context totals water, cars or charge from a rate given by a formula, a table or a graph, with the units of the situation.
Motion. Position, displacement and distance from velocity shows that integrating velocity v gives displacement, while integrating speed \lvert v \rvert gives total distance traveled.
The amount as a function. Graphing an accumulation function studies g(x) = \int_a^x f(t)\,dt as a function in its own right: it rises where f > 0, falls where f < 0, and peaks where f crosses from positive to negative.
A path through it
No child requires another, so take them in any order. Net change in context is the gentlest start, then distance and displacement, then graphing an accumulation function, which also uses the first derivative test.