CAIE A-Level Further Math A2 2.4.4 Arc Length and Surface Area Questions

Practise calculating exact arc lengths and surfaces of revolution for curves with Further Mathematics Paper 2 questions and mark schemes.

Exam points

Calculate exact arc lengths and surfaces of revolution for curves.

[Maximum number: 5]

The curve C has parametric equations

x=\frac{2}{3} t^{\frac{3}{2}}-2 t^{\frac{1}{2}}, \quad y=2 t+5, \quad \text { for } 0<t \leqslant 3 .

Find the exact length of C.

$\dot{x}=t^{\frac{1}{2}}-t^{-\frac{1}{2}}, \quad \dot{y}=2$
Differentiates x and y with respect to t.

$\dot{x}^{2}+\dot{y}^{2}=\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)^{2}+4=t+2+t^{-1}=\left(t^{\frac{1}{2}}+t^{-\frac{1}{2}}\right)^{2}$
Factorises $\dot{x}^{2}+\dot{y}^{2}$ .

$\int_{0}^{3} t^{\frac{1}{2}}+t^{-\frac{1}{2}} \mathrm{~d} t=\left[\frac{2}{3} t^{\frac{3}{2}}+2 t^{\frac{1}{2}}\right]_{0}^{3}=4 \sqrt{3}$
Applies correct formula for arc length.
M1 for their $\int_{0}^{3} \sqrt{\dot{x}^{2}+\dot{y}^{2}} \mathrm{~d} t$ .
Answer must be simplified to $4 \sqrt{3}$ for A1.

5

All question bank results loaded

CAIE A-Level Further Math A2 2.4.4 Arc Length and Surface Area Questions question 1