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Accumulation Function
AP Calculus AB · Unit 6 Integration
Learning Objectives
Understand F(x) = ∫ₐˣ f(t) dt as accumulated area
Apply FTC Part 1: F′(x) = f(x)
See when F is increasing (f > 0) vs decreasing (f < 0)
Identify extrema of F where f changes sign
FTC Part 1
If F(x) = ∫ₐˣ f(t) dt, then:
F′(x) = f(x)
F increases when f > 0 (area growing)
F decreases when f < 0 (area shrinking)
F has max/min where f = 0 and changes sign
Tags
accumulation
FTC
net area
antiderivative
INTEGRAND f(t)
f(t) = t² − 4 (a=0)
f(t) = sin(t) (a=0)
f(t) = t − 2 (a=0)
f(t) = cos(t) (a=0)
Upper limit x
2.0
F(x) = ∫ₐˣ
—
f(x) = F′(x)
—
The
green shaded area
above the x-axis adds to F(x).
The
red shaded area
below subtracts from F(x).
F(x) = net signed area from a to x.