Arc length and surface area of revolutionEdexcel International A Level Further Maths: Mind map

Exam questions on Arc length and surface area of revolution

1. The curve $C$ has equation $y=\frac23x^{\frac32}$ for $x\ge0$.
   Find the exact length of the arc of $C$ from $x=3$ to $x=8$.2 marks
2. A curve $C$ has parametric equations $x=3t^2$, $y=2t^3$ for $t\ge0$.
   Find the length of the arc of $C$ from $t=0$ to $t=\sqrt3$.2 marks
3. The curve $C$ has equation $y=x^3$ for $0\le x\le1$. The arc of $C$ is rotated through $2\pi$ radians about the $x$-axis to form a surface.
   Show that the area of the surface is $S=2\pi\int_0^1x^3\sqrt{1+9x^4}\,dx$.3 marks