Integration and Accumulation of Change

Unit Overview

This unit introduces the integral, the second pillar of calculus. Where the derivative measures instantaneous rate, the integral measures accumulated change — and, geometrically, area under a curve. You will build the integral from Riemann sums, connect it to antiderivatives through the Fundamental Theorem of Calculus (FTC), and develop core techniques (basic antiderivatives, properties of definite integrals, and u-substitution).

The FTC is the crown jewel: it ties differentiation and integration together as inverse processes. This unit is one of the largest on the exam.

Big picture: Integrating a rate gives the total (accumulated) change; the FTC lets you compute integrals using antiderivatives.

Learning Objectives

• Approximate area/accumulation with left, right, and midpoint Riemann sums (and the Trapezoidal Rule).
• Express a definite integral as a limit of Riemann sums.
• State and apply both parts of the Fundamental Theorem of Calculus.
• Use properties of definite integrals (additivity, constant multiple, reversing limits).
• Find antiderivatives / indefinite integrals of basic functions.
• Evaluate integrals using u-substitution and simple algebraic techniques.
• Interpret accumulation functions g(x) = ∫ₐˣ f(t) dt.

Prerequisite Concepts

• Differentiation (to recognize antiderivatives) and the Chain Rule (for u-substitution).
• Summation and area of basic regions.
• Domain awareness (ln|x| for 1/x, etc.).

Key Definitions

• Antiderivative: F is an antiderivative of f if F'(x) = f(x). The indefinite integral is ∫ f(x) dx = F(x) + C.
• Definite integral: ∫ₐᵇ f(x) dx = the limit of Riemann sums = signed area between f and the x-axis on [a, b].
• Riemann sum: Σ f(xᵢ*) Δx over subintervals; using left, right, or midpoint sample points.
• Accumulation function: g(x) = ∫ₐˣ f(t) dt — total accumulated amount from a to x.

Important Theorems

Fundamental Theorem of Calculus, Part 1 (FTC-1): If g(x) = ∫ₐˣ f(t) dt with f continuous, then g'(x) = f(x). With a variable upper limit u(x): d/dx ∫ₐ^{u(x)} f(t) dt = f(u(x))·u'(x) (Chain Rule).

Fundamental Theorem of Calculus, Part 2 (FTC-2): If F is any antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a).

Riemann-sum over/under-estimates: for an increasing function, left sums underestimate and right sums overestimate (reverse for decreasing); for a concave-up function, the Trapezoidal Rule overestimates.

Important Formulas

Basic antiderivatives (add + C):

• ∫ x^n dx = x^(n+1)/(n+1) for n ≠ −1; ∫ (1/x) dx = ln|x|
• ∫ e^x dx = e^x; ∫ e^(kx) dx = (1/k)e^(kx)
• ∫ cos x dx = sin x; ∫ sin x dx = −cos x
• ∫ sec² x dx = tan x
• ∫ 1/(1 + x²) dx = arctan x; ∫ 1/√(1 − x²) dx = arcsin x

Properties of definite integrals:

• ∫ₐᵃ f = 0; ∫ₐᵇ f = −∫ᵦᵃ f
• ∫ₐᵇ f + ∫ᵦᶜ f = ∫ₐᶜ f (additivity)
• ∫ₐᵇ [k·f] = k∫ₐᵇ f; ∫ₐᵇ (f ± g) = ∫ₐᵇ f ± ∫ₐᵇ g (linearity)
• Even function: ∫₋ₐᵃ f = 2∫₀ᵃ f; odd function: ∫₋ₐᵃ f = 0

Concept Explanations

Accumulation (6.1, 6.5). Integrating a rate of change gives net change: ∫ₐᵇ f'(t) dt = f(b) − f(a). In context, "amount at time b" = "starting amount" + "accumulated change": f(b) = f(a) + ∫ₐᵇ f'(t) dt.

Riemann sums (6.2–6.3). Split [a, b] into n pieces of width Δx = (b − a)/n, sample the height at left/right/midpoints, and sum height × width. As n → ∞, the sum → the definite integral. Know which sampling over- or under-estimates based on monotonicity.

The FTC (6.4, 6.7). Part 2 turns area into arithmetic: find an antiderivative, evaluate at the bounds, subtract. Part 1 says differentiating an accumulation function recovers the integrand — the reason differentiation and integration are inverse operations.

u-Substitution (6.9). The reverse of the Chain Rule. Choose u to be an "inside" function whose derivative also appears (up to a constant). Substitute du, integrate in u, then convert back — or, for definite integrals, change the limits to u-values.

Algebraic set-up (6.10–6.11). Before integrating, simplify: split a fraction term-by-term, do long division on improper rational functions, or rewrite x²/(x²+1) = 1 − 1/(x²+1) so standard forms appear (leading to arctan, etc.).

Worked Examples

Example 1 (FTC-2). ∫₁³ x² dx = [x³/3]₁³ = 27/3 − 1/3 = 26/3.

Example 2 (Right Riemann sum). Estimate ∫₀⁴ f dx with 2 right rectangles, f(2) = 3, f(4) = 5: 2(3) + 2(5) = 16.

Example 3 (u-substitution). ∫ 2x(x² + 1)³ dx: let u = x² + 1, du = 2x dx → ∫ u³ du = u⁴/4 + C = (x² + 1)⁴/4 + C.

Example 4 (FTC-1 with Chain Rule). d/dx ∫₀^{x²} cos t dt = cos(x²)·2x = 2x cos(x²).

Example 5 (Properties). If ∫₀² f = 3 and ∫₂⁵ f = 7, then ∫₀⁵ f = 10; and ∫₀⁵ 5f = 50.

Example 6 (Accumulation in context). A bacteria culture grows at rate 3t (thousands/hour) from 5 thousand: amount after 2 h = 5 + ∫₀² 3t dt = 5 + [1.5t²]₀² = 5 + 6 = 11 thousand.

Example 7 (Absolute value). ∫₀² |x − 1| dx = ∫₀¹(1 − x)dx + ∫₁²(x − 1)dx = 1/2 + 1/2 = 1.

Graphical Interpretation

• A definite integral is signed area: area above the axis is positive, below is negative.
• The accumulation function g(x) = ∫ₐˣ f: g increases where f > 0, decreases where f < 0, has a local max where f changes + → −, and is concave up where f is increasing.
• Riemann rectangles visibly over/under-shoot depending on where the height is sampled.

Common Mistakes

• Forgetting + C on indefinite integrals.
• Sign error with ∫ sin x dx = −cos x (and the trig antiderivatives generally).
• Not changing limits (or not converting back) in u-substitution.
• Treating ∫ (1/x) dx as a power-rule case — it is ln|x|.
• Mismatching Riemann over/under estimates — depends on increasing/decreasing (and concavity for trapezoids).
• Dropping the u'(x) factor when differentiating an accumulation function with a non-x upper limit.

AP Exam Tips

• FTC-2 is your workhorse for exact definite integrals; FTC-1 for derivatives of accumulation functions (watch the Chain Rule).
• On accumulation free-response, write "amount = initial + ∫ rate" explicitly.
• For estimates, state whether a sum is an over- or under-estimate and justify with monotonicity/concavity.
• Recognize u-substitution when a factor is (a constant multiple of) the derivative of another part.
• Use symmetry (odd/even) to shortcut integrals over symmetric intervals.

Unit Summary

• The definite integral = limit of Riemann sums = signed area; integrating a rate gives accumulated change.
• FTC-1: d/dx ∫ₐˣ f = f(x); FTC-2: ∫ₐᵇ f = F(b) − F(a).
• Master basic antiderivatives, properties (additivity, linearity, symmetry), and u-substitution.
• Accumulation functions inherit behavior from f via g' = f, g'' = f'.

Next unit: solving differential equations — recovering a function from its rate of change.