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.
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.
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.
Antidifferentiation reverses derivative rules and includes an arbitrary constant. The power rule for integration excludes exponent −1, 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 ∫csc2xdx=−cotx+C and ∫cscxcotxdx=−cscx+C. For an inverse-sine pattern, ∫dx/a2−x2=arcsin(x/a)+C when a>0 and ∣x∣<a.
Before selecting a technique, simplify the integrand: divide a higher-degree polynomial by its denominator or complete a quadratic square. Match known derivative patterns, especially logarithms and inverse tangent. AB integration methods include substitution and algebraic rewriting; integration by parts, partial-fraction decomposition, and improper integrals belong to BC. A difficult-looking integrand may not have an elementary AB antiderivative, so a numerical definite integral may be appropriate.
