Accumulation Function: Area That Becomes a Function
Instead of freezing an integral into one final number, let the upper bound move. As b moves right, the shaded area grows and the accumulated value F(b) changes with it. The key question is how fast that total is growing at the current boundary.
Move the idea
Try it: move the upper bound
Change one parameter and watch what changes with it.
The accumulated area F(b) is a function, not one frozen number. As the boundary moves, its slope reflects the current height f(b) — the observation that prepares the Fundamental Theorem.
A moving endpoint
The region is not fixed. Each new position adds another strip of area, so F changes continuously with b.
Why the curve bends
For f(x)=x, later strips are taller than earlier strips. The accumulated total therefore grows faster and faster.
Bridge to the theorem
The rate at which accumulated area grows is tied to the current height of the original function. This lesson isolates that moving-area idea; the dedicated Fundamental Theorem lesson connects it formally to derivatives and antiderivatives.
Make the idea reusable.
The pattern this picture reveals
Define F(b)=∫₀ᵇ f(x)dx. As b moves, the accumulated-area function changes at a rate controlled by the current height f(b). The next lesson turns this visual pattern into the Fundamental Theorem.
Check the example at b=2
Here f(x)=x, so F(b)=b²/2. Differentiating gives F′(b)=b, exactly the same as f(b). At b=2 the accumulated area is 2, while its instantaneous growth rate is also 2.
Area and rate are different quantities
F(b) is the total area accumulated up to b; F′(b) is how fast that total is changing right now. The theorem connects them, but they are not generally the same number.