Arc length and surface area of revolutionEdexcel International A Level Further Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=\frac23x^{\frac32}$ for $x\ge0$.

   (a)

   Which integral gives the length of the arc of $C$ from $x=0$ to $x=3$?

   [1 mark]

   • A$\int_0^3(1+x)\,dx$
   • B$\int_0^3\sqrt{1+\sqrt{x}}\,dx$
   • C$\int_0^3\sqrt{1+x}\,dx$
   • D$\int_0^3\sqrt{1+x^2}\,dx$

   (b)

   Find the exact length of the arc of $C$ from $x=0$ to $x=3$.

   [1 mark]

   • A$\frac{21}{2}$
   • B$7$
   • C$2\sqrt3$
   • D$\frac{14}{3}$

   (c)

   Find the exact length of the arc of $C$ from $x=3$ to $x=8$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve $C$ has parametric equations $x=3t^2$, $y=2t^3$ for $t\ge0$.

   (a)

   Find $\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2$.

   [1 mark]

   • A$36t^2+36t^4$
   • B$6t+6t^2$
   • C$9t^4+4t^6$
   • D$36t^2+6t^4$

   (b)

   Find the exact length of the arc of $C$ from $t=0$ to $t=1$.

   [1 mark]

   • A$4\sqrt2$
   • B$4\sqrt2-2$
   • C$2\sqrt2-1$
   • D$9\sqrt2-\frac92$

   (c)

   Find the length of the arc of $C$ from $t=0$ to $t=\sqrt3$.

   [2 marks]

   Total for question 2: 4 marks

3. 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.

   (a)

   Show that the area of the surface is $S=2\pi\int_0^1x^3\sqrt{1+9x^4}\,dx$.

   [3 marks]

   (b)

   Find the exact value of $S$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has parametric equations $x=t^2$, $y=\frac{t^3}{3}$ for $0\le t\le\sqrt5$.

   (a)

   Show that the length of $C$ is $\frac{19}{3}$.

   [6 marks]

   (b)

   The arc $C$ is rotated through $2\pi$ radians about the $y$-axis. Find the exact area of the surface formed.

   [6 marks]

   Total for question 4: 12 marks

End of questions