Standard +0.8 This is a standard arc length formula application requiring knowledge of hyperbolic function derivatives and the identity cosh²x - sinh²x = 1. While it involves Further Maths content (hyperbolic functions), the derivation is straightforward: differentiate cosh x to get sinh x, substitute into the arc length formula to get ∫√(1 + sinh²x)dx = ∫cosh x dx, which integrates directly to sinh x. The question is routine for Further Maths students who know the key identity, making it moderately above average difficulty on an absolute scale.
A curve has equation \(y = \cosh x\)
Show that the arc length of the curve from \(x = a\) to \(x = b\), where \(0 < a < b\), is equal to
$$\sinh b - \sinh a$$
[4 marks]
A curve has equation $y = \cosh x$
Show that the arc length of the curve from $x = a$ to $x = b$, where $0 < a < b$, is equal to
$$\sinh b - \sinh a$$
[4 marks]
\hfill \mbox{\textit{AQA Further Paper 2 2019 Q5 [4]}}