Integration and Accumulation of Change

What you'll learn

This revision guide covers integration techniques and accumulation concepts essential for AP Calculus AB. You will master antiderivatives, definite and indefinite integrals, Riemann sums, the Fundamental Theorem of Calculus, and applications including area, volume, and motion problems. These topics typically account for 17-20% of the AP Calculus AB exam.

Key terms and definitions

Antiderivative — A function F(x) whose derivative is f(x); if F'(x) = f(x), then F(x) is an antiderivative of f(x)

Indefinite integral — The general antiderivative of a function, written as ∫f(x)dx = F(x) + C, where C is the constant of integration

Definite integral — The accumulation of a rate of change over an interval [a,b], written as ∫ₐᵇf(x)dx, representing the net signed area between f(x) and the x-axis

Riemann sum — An approximation of a definite integral using rectangles; can be left, right, midpoint, or trapezoidal

Fundamental Theorem of Calculus (FTC) — The theorem connecting differentiation and integration, existing in two parts: Part 1 establishes that differentiation and integration are inverse operations; Part 2 provides the evaluation method for definite integrals

Accumulation function — A function defined by A(x) = ∫ₐˣf(t)dt, representing the accumulated value from a to x

Net change — The definite integral ∫ₐᵇf'(x)dx = f(b) - f(a), representing total change in f over [a,b]

Displacement — The net change in position, calculated as ∫ₐᵇv(t)dt, distinct from total distance travelled

Core concepts

Antiderivatives and indefinite integrals

The process of finding antiderivatives reverses differentiation. Every antiderivative of f(x) differs only by a constant.

Basic antiderivative rules:

• Power rule: ∫xⁿdx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1
• ∫(1/x)dx = ln|x| + C
• ∫eˣdx = eˣ + C
• ∫sin(x)dx = -cos(x) + C
• ∫cos(x)dx = sin(x) + C
• ∫sec²(x)dx = tan(x) + C

Constant multiple rule: ∫kf(x)dx = k∫f(x)dx

Sum/difference rule: ∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx

Always include the constant of integration C when finding indefinite integrals. Initial conditions allow you to determine specific values of C.

U-substitution

U-substitution simplifies integrals by changing variables, particularly useful when the integrand contains a composite function.

Process:

1. Identify u as the inner function
2. Calculate du = u'dx
3. Rewrite the entire integral in terms of u
4. Integrate with respect to u
5. Substitute back to express the answer in terms of x

For definite integrals, either:

• Change the limits of integration to u-values, or
• Substitute back to x and use original limits

Example pattern: ∫2x(x² + 1)⁵dx

• Let u = x² + 1
• Then du = 2x dx
• Integral becomes ∫u⁵du = (u⁶/6) + C = (x² + 1)⁶/6 + C

Riemann sums and numerical integration

Riemann sums approximate definite integrals by dividing the interval into n subintervals of width Δx = (b-a)/n.

Left Riemann sum: Uses left endpoints: Δx[f(x₀) + f(x₁) + ... + f(xₙ₋₁)]

Right Riemann sum: Uses right endpoints: Δx[f(x₁) + f(x₂) + ... + f(xₙ)]

Midpoint Riemann sum: Uses midpoints of subintervals

Trapezoidal rule: Approximates using trapezoids instead of rectangles: (Δx/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]

The trapezoidal rule typically provides better approximations than rectangular methods. As n increases, all Riemann sums approach the exact value of the definite integral.

The Fundamental Theorem of Calculus

Part 1 (FTC 1): If f is continuous on [a,b] and F(x) = ∫ₐˣf(t)dt, then F'(x) = f(x).

This means the derivative of an accumulation function equals the original function. It establishes integration and differentiation as inverse operations.

Applications of FTC 1:

• Finding derivatives of functions defined by integrals
• Chain rule application: d/dx[∫ₐᵍ⁽ˣ⁾f(t)dt] = f(g(x))·g'(x)

Part 2 (FTC 2): If F is any antiderivative of f on [a,b], then: ∫ₐᵇf(x)dx = F(b) - F(a)

This provides the evaluation method for definite integrals without computing Riemann sum limits.

Notation: F(b) - F(a) is often written as [F(x)]ₐᵇ or F(x)|ₐᵇ

Properties of definite integrals

Understanding these properties helps simplify calculations and solve complex problems:

Interval additivity: ∫ₐᵇf(x)dx + ∫ᵦᶜf(x)dx = ∫ₐᶜf(x)dx

Reversal of limits: ∫ₐᵇf(x)dx = -∫ᵦₐf(x)dx

Zero-width interval: ∫ₐₐf(x)dx = 0

Constant multiple: ∫ₐᵇkf(x)dx = k∫ₐᵇf(x)dx

Sum/difference: ∫ₐᵇ[f(x) ± g(x)]dx = ∫ₐᵇf(x)dx ± ∫ₐᵇg(x)dx

Comparison: If f(x) ≥ g(x) on [a,b], then ∫ₐᵇf(x)dx ≥ ∫ₐᵇg(x)dx

Applications of integration

Area between curves:

The area between f(x) and g(x) from x = a to x = b, where f(x) ≥ g(x): Area = ∫ₐᵇ[f(x) - g(x)]dx

For vertical strips, integrate with respect to x. For horizontal strips, express functions as x = f(y) and integrate with respect to y.

Volumes of revolution:

Disk method: Rotating about a horizontal axis (e.g., x-axis): V = π∫ₐᵇ[R(x)]²dx, where R(x) is the radius function

Washer method: For regions between two curves: V = π∫ₐᵇ{[R(x)]² - [r(x)]²}dx, where R(x) is outer radius and r(x) is inner radius

Motion problems:

Given velocity v(t):

• Displacement: ∫ₐᵇv(t)dt (net change in position, can be negative)
• Total distance: ∫ₐᵇ|v(t)|dt (always positive; split integral where v(t) = 0)

Given acceleration a(t):

• Change in velocity: ∫ₐᵇa(t)dt = v(b) - v(a)

Accumulation from rates:

When f'(x) represents a rate of change, ∫ₐᵇf'(x)dx gives the total accumulation. Context examples include:

• Population growth from birth rate
• Volume change from flow rate
• Temperature change from rate of heating

Worked examples

Example 1: Definite integral with FTC

Question: Evaluate ∫₁⁴(3x² - 2x + 5)dx

Solution:

Find the antiderivative: F(x) = x³ - x² + 5x + C

Apply FTC Part 2: ∫₁⁴(3x² - 2x + 5)dx = [x³ - x² + 5x]₁⁴

Evaluate at the bounds: = [4³ - 4² + 5(4)] - [1³ - 1² + 5(1)] = [64 - 16 + 20] - [1 - 1 + 5] = 68 - 5 = 63

Answer: 63

Example 2: U-substitution with definite integral

Question: Evaluate ∫₀^(π/2) sin(x)cos³(x)dx

Solution:

Let u = cos(x) Then du = -sin(x)dx, so sin(x)dx = -du

Change limits:

• When x = 0: u = cos(0) = 1
• When x = π/2: u = cos(π/2) = 0

Substitute: ∫₀^(π/2) sin(x)cos³(x)dx = ∫₁⁰u³(-du) = -∫₁⁰u³du = ∫₀¹u³du

Integrate: = [u⁴/4]₀¹ = 1/4 - 0 = 1/4

Answer: 1/4

Example 3: Accumulation function derivative

Question: Let g(x) = ∫₂ˣ(t² - 3t)dt. Find g'(x) and g'(3).

Solution:

By FTC Part 1, the derivative of an accumulation function equals the integrand with the variable of integration replaced by the upper limit:

g'(x) = x² - 3x

Evaluate at x = 3: g'(3) = 3² - 3(3) = 9 - 9 = 0

Answer: g'(x) = x² - 3x; g'(3) = 0

Example 4: Area between curves

Question: Find the area of the region bounded by y = x² and y = 2x from x = 0 to x = 2.

Solution:

First, identify which function is on top. Test at x = 1:

• y = 2x gives y = 2
• y = x² gives y = 1

So 2x ≥ x² on this interval.

Area = ∫₀²(2x - x²)dx

Find antiderivative: = [x² - x³/3]₀²

Evaluate: = [4 - 8/3] - [0] = 12/3 - 8/3 = 4/3

Answer: 4/3 square units

Common mistakes and how to avoid them

• Forgetting the constant of integration — Always include "+ C" for indefinite integrals. Definite integrals do not need C because it cancels when evaluating F(b) - F(a).

• Incorrectly applying the power rule when n = -1 — Remember ∫x⁻¹dx = ln|x| + C, not x⁰/0. The power rule does not apply to n = -1.

• Mixing up displacement and distance — Displacement is ∫v(t)dt (can be negative); total distance requires ∫|v(t)|dt and splitting the integral where velocity changes sign.

• Changing only part of the integral during u-substitution — When substituting u, ensure dx is also converted to du. For definite integrals, either change both limits or substitute back to x.

• Reversing the order in subtraction for areas — Always subtract the lower function from the upper function: ∫[top - bottom]dx. Sketch the region if uncertain.

• Forgetting to apply chain rule with FTC 1 — For d/dx[∫ₐᵍ⁽ˣ⁾f(t)dt], multiply f(g(x)) by g'(x). Also handle variable lower limits: d/dx[∫ₕ₍ₓ₎ᵇf(t)dt] = -f(h(x))·h'(x).

Exam technique for "Integration and Accumulation of Change"

• Show all integration steps clearly — Write the antiderivative, then evaluate at bounds using bracket notation [F(x)]ₐᵇ. This earns partial credit even if arithmetic errors occur later.

• Identify keywords in application problems — "Net change" signals a direct definite integral; "total amount" or "distance travelled" may require absolute values or splitting integrals at zeros.

• Use correct units in context — If velocity is in m/s and time in seconds, displacement has units of metres. Always state units in real-world problems for full marks.

• Check reasonableness of numerical answers — If calculating area and obtaining a negative value, revisit your function order. Riemann sum approximations should be close to exact integral values; large discrepancies indicate errors.

Quick revision summary

Integration reverses differentiation. Indefinite integrals include "+ C"; definite integrals use FTC Part 2: ∫ₐᵇf(x)dx = F(b) - F(a). U-substitution handles composite functions. Riemann sums approximate integrals numerically. FTC Part 1 states d/dx[∫ₐˣf(t)dt] = f(x). Applications include area between curves (integrate top minus bottom), volumes of revolution (disk/washer methods), and motion (displacement vs. distance). Always sketch regions, show work systematically, and verify answers contextually.