Summary AP Calculus AB Unit 6: Integration and Accumulation of Change - Study Guide

AP Calculus AB Unit 6: Integration & Accumulation



Integration
Accumulation of Change and Antiderivatives




1. Approximating Area (Riemann Sums)

We approximate the area under a curve f (x) on [a, b] using rectangles.
n
X b−a
Area ≈ f (ci )∆x, where ∆x =
n
i=1


• Left Riemann Sum (LRAM): Use left endpoint of subinterval.

• Right Riemann Sum (RRAM): Use right endpoint of subinterval.

• Midpoint Riemann Sum (MRAM): Use midpoint.

• Trapezoidal Sum: Average of Left and Right (more accurate).

∆x
Area ≈ [f (x0 ) + 2f (x1 ) + 2f (x2 ) + · · · + f (xn )]
2

Over/Under Estimation:

• Increasing Function: LRAM < Area < RRAM

• Decreasing Function: RRAM < Area < LRAM


2. Definite Integrals as Limits

The exact area is the limit of the Riemann Sums as n → ∞.

Definition of Definite Integral
Z b n
X
f (x) dx = lim f (ci )∆x
a n→∞
i=1

Represents net signed area between the graph and the x-axis. Area below the axis counts as
negative.



3. Antiderivatives & Indefinite Integrals

Finding the function F (x) such that F ′ (x) = f (x). Don’t forget +C!




1