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LearnAQA7367First teaching 2017Paper 2 · Compulsory contentFurther calculus

Paper 2 · Module

Further calculus

3 subtopics≈ 135 min in total

Start the module: Volumes, mean values and improper integrals Open graphing calculator

Volumes, mean values and improper integrals
Spin a curve about an axis and the solid's volume is $\pi\int y^2\,dx$ (or $\pi\int x^2\,dy$ ). The mean value of $f$ on $[a,b]$ is $\frac1{b-a}\int_a^bf\,dx$ . An improper integral has an infinite limit or an infinite integrand: replace it by a limit and see whether that limit exists.
≈ 45 min
Inverse trig and hyperbolic functions in calculus
Differentiating $y=\arcsin x$ implicitly gives $\frac1{\sqrt{1-x^2}}$ ; similar results hold for $\arctan$ , $\operatorname{arsinh}$ , $\operatorname{arcosh}$ . Read backwards, these give the integrals of $\frac1{\sqrt{a^2-x^2}}$ , $\frac1{a^2+x^2}$ and $\frac1{\sqrt{x^2\pm a^2}}$ , which algebra alone can't do.
≈ 45 min
Maclaurin series
Any smooth function can be approximated near $0$ by a polynomial whose derivatives match it there: $f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\cdots$ . Standard series for $e^x$ , $\sin x$ , $\cos x$ , $\ln(1+x)$ and $(1+x)^n$ combine by substitution, multiplication and differentiation.
≈ 45 min