Code Scholars Tutoring · AP Calculus BC
14 topics across 9 lessons, with explanations, worked examples, and self-checks that develop complete mathematical reasoning.
Topics 6.1, 6.2, 6.3
Approximate a changing quantity by summing rate times interval width. Left, right, and midpoint sums use different sample locations. Trapezoids average endpoint heights; nonuniform partitions require their individual widths. A definite integral is a limit of signed sums, so portions below the axis subtract. For an increasing function, left sums underestimate and right sums overestimate; concavity controls trapezoidal error.
Topics 6.4, 6.5
When the integrand is continuous, an accumulation function differentiates to its integrand at the upper limit. A moving upper limit adds a chain-rule factor. Moving lower and upper limits produce two terms. Determine increase, extrema, and concavity of an accumulation function by analyzing its derivative, not by trying to find an elementary antiderivative first.
Topics 6.6, 6.7
Definite integrals add across adjacent intervals and reverse sign when bounds are exchanged. Linearity allows constants to factor out and sums to split. Evaluate using an antiderivative when available. Symmetry can simplify work: odd integrands have zero signed integral on symmetric bounds, while even integrands give twice the positive-half integral. State continuity or other sufficient conditions when using the FTC.
Topics 6.8, 6.9
Antidifferentiation reverses derivative rules and includes an arbitrary constant. The power rule for integration excludes exponent , which produces a logarithm. Substitution matches a composite expression with its derivative. For definite integrals either change both bounds into the new variable or substitute back before evaluating; never mix transformed variables with original bounds. Other useful reverse patterns are and . For an inverse-sine pattern, when and .
Topics 6.10
Algebra can expose an antiderivative before a more elaborate technique is needed. Compare polynomial degrees and divide when the numerator degree is at least the denominator degree. Complete a quadratic square when it reveals an inverse-tangent pattern. Keep track of the original domain while rewriting the expression. In BC, this algebraic preparation also helps decide whether substitution, integration by parts, or linear partial fractions is the most useful next step.
Topics 6.11 · BC only
Integration by parts reverses the product rule. It is useful when an integrand contains a product whose factors play different roles: differentiating one factor simplifies it, while the other has an antiderivative you can find. The goal is to replace the original integral with an easier one, not simply to make it look different.
Topics 6.12 · BC only
Partial fractions rewrites a rational function as a sum of simpler rational functions. When the denominator factors into distinct linear factors, each resulting term has a constant numerator and a logarithmic antiderivative. The decomposition is an algebraic identity on the original function's domain.
Topics 6.13 · BC only
An improper integral involves an infinite interval or an unbounded integrand. Its value is defined by a limit of ordinary definite integrals. A finite limit means the improper integral converges; an infinite or nonexistent limit means it diverges. An antiderivative alone does not settle convergence.
Topics 6.14
Method selection starts with the structure of the integrand and the conditions of the problem. There is no single integration rule that solves every expression. In BC, algebraic rewriting and substitution sit alongside integration by parts, linear partial fractions, and limits for improper integrals.
