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Unit 6: Integration and Accumulation of Change

13 dot points across 13 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

What algebraic properties let you split, combine and reverse definite integrals?

Topic 6.6 Applying Properties of Definite Integrals: use linearity, additivity over intervals, and limit-reversal properties of definite integrals.
A focused answer to AP Calculus AB Topic 6.6, applying the linearity, interval-additivity, and limit-reversal properties of definite integrals to combine and manipulate given integral values, with worked examples.
8 min answer →

How do left, right, midpoint and trapezoidal sums approximate the area under a curve, and when do they over- or under-estimate?

Topic 6.2 Approximating Areas with Riemann Sums: approximate area using left, right, midpoint, and trapezoidal sums, and reason about over- and under-estimates.
A focused answer to AP Calculus AB Topic 6.2, approximating area under a curve with left, right, midpoint, and trapezoidal sums, with worked table-based computations and reasoning about over- and under-estimation.
10 min answer →

How do you evaluate an integral with an infinite limit or an unbounded integrand using limits?

Topic 6.13 Evaluating Improper Integrals: evaluate integrals with infinite limits of integration or an infinite discontinuity by rewriting them as limits of proper integrals, determining convergence or divergence (BC).
A focused answer to AP Calculus BC Topic 6.13, evaluating improper integrals with infinite limits or unbounded integrands by replacing the bad endpoint with a limit, and deciding convergence versus divergence, with worked examples.
10 min answer →

How does the area under a rate-of-change graph represent the accumulated change in a quantity?

Topic 6.1 Exploring Accumulations of Change: interpret the area under a rate graph as the net accumulated change in a quantity.
A focused answer to AP Calculus AB Topic 6.1, interpreting the area under a rate-of-change graph as net accumulated change, including signed area and units, with worked geometric-area examples.
8 min answer →

How do you find indefinite integrals of basic functions by reversing the differentiation rules?

Topic 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation: find indefinite integrals of power, trigonometric, exponential and reciprocal functions.
A focused answer to AP Calculus AB Topic 6.8, finding indefinite integrals of power, exponential, reciprocal and trigonometric functions by reversing the derivative rules, with the constant of integration and worked examples.
9 min answer →

How does integration by parts reverse the product rule to integrate a product of functions?

Topic 6.11 Integrating Using Integration by Parts: integrate products of functions by reversing the product rule, choosing the parts and applying the formula, including repeated use (BC).
A focused answer to AP Calculus BC Topic 6.11, integrating products of functions by reversing the product rule, choosing u and dv with LIATE, and applying integration by parts including repeated use, with worked examples.
10 min answer →

How does u-substitution reverse the chain rule to integrate composite functions?

Topic 6.9 Integrating Using Substitution: integrate composite functions by reversing the chain rule with u-substitution, including changing limits for definite integrals.
A focused answer to AP Calculus AB Topic 6.9, integrating composite functions by u-substitution as the reverse of the chain rule, including changing the limits of definite integrals, with worked examples.
10 min answer →

How do you analyze the increasing, decreasing and concavity behavior of an accumulation function from the graph of its integrand?

Topic 6.5 Interpreting the Behavior of Accumulation Functions Involving Area: analyze extrema and concavity of an accumulation function using the graph of the integrand.
A focused answer to AP Calculus AB Topic 6.5, analyzing the increasing/decreasing, extrema, and concavity behavior of an accumulation function from the graph of its integrand, with worked area-based reasoning.
9 min answer →

How does the limit of a Riemann sum become the definite integral, and what does summation and integral notation mean?

Topic 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation: express a Riemann sum in summation notation and define the definite integral as its limit.
A focused answer to AP Calculus AB Topic 6.3, expressing Riemann sums in summation notation and defining the definite integral as the limit of Riemann sums, with worked translations between sum and integral notation.
9 min answer →

How do you choose the right antidifferentiation technique for a given integral?

Topic 6.14 Selecting Techniques for Antidifferentiation: choose between rewriting, basic rules, and substitution to evaluate an integral.
A focused answer to AP Calculus AB Topic 6.14, choosing among algebraic rewriting, basic antiderivative rules, and u-substitution for a given integral, with worked decision examples for the AB toolkit.
9 min answer →

What is an accumulation function, and how does the Fundamental Theorem of Calculus give its derivative?

Topic 6.4 The Fundamental Theorem of Calculus and Accumulation Functions: differentiate accumulation functions using the first part of the Fundamental Theorem of Calculus.
A focused answer to AP Calculus AB Topic 6.4, defining accumulation functions and using the first part of the Fundamental Theorem of Calculus, with the chain rule for variable upper limits, in worked examples.
9 min answer →

How does the Fundamental Theorem of Calculus let you evaluate a definite integral using an antiderivative?

Topic 6.7 The Fundamental Theorem of Calculus and Definite Integrals: evaluate definite integrals using the second part of the Fundamental Theorem of Calculus.
A focused answer to AP Calculus AB Topic 6.7, evaluating definite integrals with the second part of the Fundamental Theorem of Calculus by finding an antiderivative and computing the difference at the limits, with worked examples.
9 min answer →

How do you integrate a rational function by splitting it into linear partial fractions?

Topic 6.12 Using Linear Partial Fractions: rewrite a rational function with distinct linear factors in the denominator as a sum of partial fractions and integrate each to a logarithm (BC).
A focused answer to AP Calculus BC Topic 6.12, decomposing a rational function with distinct linear denominator factors into partial fractions and integrating each piece to a natural logarithm, with worked examples.
9 min answer →