Arc length with substitution

A question is this type if and only if it requires finding an arc length where the integrand does not simplify trivially and an explicit substitution (such as hyperbolic or algebraic) is needed or suggested to evaluate the resulting integral.

2 questions · Challenging +1.8

Sort by: Default | Easiest first | Hardest first
Edexcel F3 2014 June Q7
13 marks Challenging +1.8
7. The curve \(C\) has parametric equations $$x = 3 t ^ { 2 } , \quad y = 12 t , \quad 0 \leqslant t \leqslant 4$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  1. Show that the area of the surface generated is $$\pi ( a \sqrt { 5 } + b )$$ where \(a\) and \(b\) are constants to be found.
  2. Show that the length of the curve \(C\) is given by $$k \int _ { 0 } ^ { 4 } \sqrt { \left( t ^ { 2 } + 4 \right) } \mathrm { d } t$$ where \(k\) is a constant to be found.
  3. Use the substitution \(t = 2 \sinh \theta\) to show that the exact value of the length of the curve \(C\) is $$24 \sqrt { 5 } + 12 \ln ( 2 + \sqrt { 5 } )$$
Edexcel FP3 2009 June Q8
11 marks Challenging +1.8
  1. A curve, which is part of an ellipse, has parametric equations
$$x = 3 \cos \theta , \quad y = 5 \sin \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 } .$$ The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
  1. Show that the area of the surface generated is given by the integral $$k \pi \int _ { 0 } ^ { \alpha } \sqrt { } \left( 16 c ^ { 2 } + 9 \right) \mathrm { d } c , \quad \text { where } c = \cos \theta$$ and where \(k\) and \(\alpha\) are constants to be found.
  2. Using the substitution \(c = \frac { 3 } { 4 } \sinh u\), or otherwise, evaluate the integral, showing all of your working and giving the final answer to 3 significant figures.