AP Calculus: Integration and Accumulation of Change

This chapter introduces integration, the process of finding the area under a curve, which represents the accumulation of a quantity. You will begin by approximating this area using rectangles and trapezoids in what are called Riemann sums. These approximations form the basis for understanding the definite integral, which gives the exact area or net change as the limit of these sums.

The central concept is the Fundamental Theorem of Calculus, which connects integration back to differentiation. You will use this theorem to evaluate integrals by finding antiderivatives and to differentiate functions that are defined by an integral. The primary technique for finding more complex antiderivatives is u-substitution. Applications include calculating the area between curves and solving problems where you are given a rate of change and need to find the total amount accumulated over an interval.

Students often find the conceptual jump from Riemann sums to the definite integral challenging, as well as correctly applying the second part of the Fundamental Theorem. Setting up integrals for application problems requires careful thought. This chapter relies heavily on a strong understanding of differentiation from the previous unit.

Formulas for AP Calculus: Integration and Accumulation of Change

Power Rule for Integration: $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$ The basic rule for finding the antiderivative of a power function, for n ≠ -1.
The Definite Integral as a Limit of a Riemann Sum: $\int_{a}^{b} f(x) \,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$ The formal definition of the definite integral, representing the exact area under the curve.
Fundamental Theorem of Calculus, Part 1 (Evaluation): $\int_{a}^{b} f(x) \,dx = F(b) - F(a)$ Where F is an antiderivative of f, so F'(x) = f(x).
Accumulation Function (Net Change Theorem): $F(b) = F(a) + \int_{a}^{b} F'(t) \,dt$ Calculates the final amount given an initial amount and a rate of change.
Fundamental Theorem of Calculus, Part 2 (Derivatives of Integrals): $\frac{d}{dx} \int_{a}^{x} f(t) \,dt = f(x)$ Shows that differentiation and integration are inverse operations.
FTC Part 2 with the Chain Rule: $\frac{d}{dx} \int_{a}^{g(x)} f(t) \,dt = f(g(x)) \cdot g'(x)$ Used when the upper limit of integration is a function of x.
Integration by Substitution (u-substitution): $\int f(g(x))g'(x) \,dx = \int f(u) \,du$ The primary method for reversing the chain rule, where u = g(x).
Average Value of a Function: $f_{avg} = \frac{1}{b-a} \int_{a}^{b} f(x) \,dx$ Finds the mean value of the function f(x) over the interval [a, b].
Area Between Two Curves: $A = \int_{a}^{b} (f(x) - g(x)) \,dx$ Calculates the area where f(x) is the upper curve and g(x) is the lower curve.