AP Calculus AB Unit 6 Practice Test: Integration and Accumulation of Change

From Differentiation to Integration

Unit 6 introduces the second major branch of calculus: integration. Where differentiation analyzes instantaneous rates of change, integration accumulates total change over an interval. The Fundamental Theorem of Calculus is the bridge between these two ideas — one of the most important mathematical results you will encounter in AP Calculus AB. This unit test covers the full range of integration concepts that appear on the AP exam.

Topics in This Unit Practice Test

• Riemann sums — left, right, midpoint, and trapezoidal approximations
• The definite integral as a limit of Riemann sums
• Properties of definite integrals
• The Fundamental Theorem of Calculus, Part 1: d/dx ∫[a to x] f(t) dt = f(x)
• The Fundamental Theorem of Calculus, Part 2: ∫[a to b] f(x) dx = F(b) − F(a)
• Antiderivatives and indefinite integrals
• Basic integration rules: power rule, constant rule, trig integrals
• The technique of u-substitution

The Fundamental Theorem of Calculus on the AP Exam

FTC Part 1 is frequently tested in accumulation function problems where students must differentiate an integral with a variable upper limit. The chain rule version — when the upper limit is a function of x — appears in both MCQ and FRQ. FTC Part 2 is the basis for evaluating all definite integrals using antiderivatives. Understanding the distinction between the two parts, and when to apply each, is a high-value AP skill.

U-Substitution: The Core Integration Technique

U-substitution is the integration counterpart of the chain rule. It is used when an integrand contains a composite structure — an inner function and its derivative (or a scalar multiple of it). Success with u-substitution depends on correctly identifying the substitution, adjusting the differential (dx to du), transforming the integral, integrating in terms of u, and back-substituting. For definite integrals, you can either back-substitute and evaluate, or change the limits of integration in terms of u.

Connecting Integration and Differentiation

Unit 6 is where the relationship between derivatives and integrals becomes explicit. An antiderivative undoes differentiation; the definite integral gives the net signed area under a curve. These connections are foundational for Units 7 and 8, where integration is applied to differential equations and real-world accumulation problems. Students who understand Unit 6 conceptually — not just procedurally — are far better equipped for the AP exam's FRQ section.