Surface area with arc length identity

A question is this type if and only if it requires first establishing that (dx/dt)² + (dy/dt)² simplifies to a perfect square (shown explicitly), and then uses this to compute arc length or surface area.

2 questions · Challenging +1.5

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OCR MEI FP3 2008 June Q3
24 marks Challenging +1.2
3 The curve \(C\) has parametric equations \(x = 8 t ^ { 3 } , y = 9 t ^ { 2 } - 2 t ^ { 4 }\), for \(t \geqslant 0\).
  1. Show that \(\dot { x } ^ { 2 } + \dot { y } ^ { 2 } = \left( 18 t + 8 t ^ { 3 } \right) ^ { 2 }\). Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 2\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  3. Show that the curvature at a general point on \(C\) is \(\frac { - 6 } { t \left( 4 t ^ { 2 } + 9 \right) ^ { 2 } }\).
  4. Find the coordinates of the centre of curvature corresponding to the point on \(C\) where \(t = 1\).
AQA Further Paper 1 2023 June Q16
11 marks Challenging +1.8
  1. Show that $$\int_{0.5}^4 \frac{1}{t} \ln t \, \mathrm{d}t = a(\ln 2)^2$$ where \(a\) is a rational number to be found. [4 marks]
  2. A curve C is defined parametrically for \(t > 0\) by $$x = 2t \quad y = \frac{1}{2}t^2 - \ln t$$ The arc formed by the graph of C from \(t = 0.5\) to \(t = 4\) is rotated through \(2\pi\) radians about the \(x\)-axis to generate a surface with area \(S\) Find the exact value of \(S\), giving your answer in the form $$S = \pi\left(b + c \ln 2 + d(\ln 2)^2\right)$$ where \(b\), \(c\) and \(d\) are rational numbers to be found. [7 marks]