Integral calculus
Chapter 50 of 16 strong
RESTS ONConstruct inverse circular derivatives,exact gradients,domains and tangent/normal linesCH 4State the complete anti-derivative family with one arbitrary additive constant, and verify that the constant disappears under differentiationFOUNDATIONSGiven d/dx[F(x)] = f(x), conclude ∫ f(x) dx = F(x) + c, and use that derivative to anti-differentiate a stated exact multiple of f(x), matching the chain-rule factor rather than integrating the visible factors separatelyFOUNDATIONSUse exact linearity, interval reversal, interval splitting, a zero-width interval and invariance under renaming the dummy variableFOUNDATIONSConstruct direct, contrapositive and contradiction arguments with justified stepsCH 1Construct exact rational decompositions with complete real factor patterns and domain restrictionsCH 2Apply exact compound and double-angle identities with valid domainsFOUNDATIONSIdentify two factors, differentiate each, combine them as u'v + uv' in the stated order, and simplify or refactor the result only where the question asks for a factorised formFOUNDATIONSSelect and evaluate a definite integral for a region bounded by a low-degree polynomial, the horizontal axis and stated vertical boundaries, splitting at constructed axis crossings when actual area is requiredFOUNDATIONSRearrange a polynomial equation to zero, factorise it, and use the null factor law to obtain all real solutionsFOUNDATIONSApproximate the area between a curve and the horizontal axis over a stated interval by the trapezium rule: partition the interval into equal-width subintervals, evaluate the integrand at the endpoints of each, and combine the heights with the trapezoidal weighting (w/2)(h_0 + 2h_1 + ... + 2h_(n−1) + h_n) for the common width w, rather than the rectangle weightingFOUNDATIONSInterpret an integral as signed area relative to the horizontal axis and distinguish that value from non-negative geometric area when the graph is below or crosses the axisFOUNDATIONSRoundingFOUNDATIONSFind first/second parametric derivatives and classify regular or singular tangentsCH 7Differentiate and integrate vector motion with speed and initial conditionsCH 7
Inverse circular antiderivatives and substitution
Partial fractions, trigonometric identities and integration by parts
Areas and numerical/symbolic integration
Volumes, parametric lengths and surface areas