AP Calculus AB · Unit 6 Integration & Accumulation
Learning Objectives
The Accumulation Function: $F(x) = \int_a^x f(t)dt$. As $x$ increases, the integral "sweeps" across the graph, sweeping up exact area.
FTC Part I: The derivative of the accumulation function is simply the original function inside! $\frac{d}{dx}\int_a^x f(t)dt = f(x)$.
Negative Area: When $f(t)$ dips below the x-axis, the accumulation function $F(x)$ starts losing value (subtracting area).
Tags
IntegrationFundamental TheoremAccumulation
DRAG THE X-BOUNDARY TO ACCUMULATE AREA
Integrand f(t)
Accumulation F(x)
Select Integrand (Velocity)
Evaluation Box
$F(x) = \int_{0}^{x} f(t) dt$
CURRENT UPPER BOUND
Evaluate at $x =$0.00
$f(x)$ (Rate) =$0.00
TOTAL ACCUMULATED AREA
$F(x) =$0.00
Observations: Notice that when $f(t)$ is POSITIVE (above axis), $F(x)$ is INCREASING. When $f(t)$ crosses the axis and becomes NEGATIVE, $F(x)$ immediately starts DECREASING! The x-intercepts of $f(t)$ are the relative extrema of $F(x)$!