Reduction formulae, arc length, surface area and bounds for sums

Learning objectives

Derive a reduction formula for a definite integral such as ∫xⁿeˣ dx or ∫sinⁿx dx by integration by parts, stating the values of n for which it holds, and apply it repeatedly down to a base case that can be evaluated directly

Evaluate integrals by trigonometric or hyperbolic substitution, and recognise the standard forms that integrate to arcsin, arctan, arsinh and arcosh, completing the square first where necessary

Calculate arc length using ∫√(1 + (dy/dx)²) dx for a Cartesian curve, ∫√((dx/dt)² + (dy/dt)²) dt for a parametric curve and ∫√(r² + (dr/dθ)²) dθ for a polar curve

Calculate the area of a surface of revolution about the x-axis using 2π∫y√(1 + (dy/dx)²) dx, and use the corresponding parametric form (surface areas for polar curves are not required)

Estimate the area under a curve with a row of rectangles, and compare a sum such as Σ1/r² or Σ1/r with the integral of the same function to bound the sum from above or below, or to show that a series has no sum to infinity