Arc length and surface area of revolutionAQA A-Level Further Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=\frac23x^{3/2}$ for $0\le x\le3$.

   (a)

   Find $1+\left(\frac{dy}{dx}\right)^2$.

   [1 mark]

   • A$1+\sqrt x$
   • B$1+x$
   • C$\sqrt{1+x}$
   • D$1+\frac49x^3$

   (b)

   Which integral gives the length of $C$?

   [1 mark]

   • A$\int_0^3(1+x)\,dx$
   • B$\int_0^3\sqrt{1+x^2}\,dx$
   • C$2\pi\int_0^3\frac23x^{3/2}\sqrt{1+x}\,dx$
   • D$\int_0^3\sqrt{1+x}\,dx$

   (c)

   Find the exact length of $C$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=\sqrt x$ for $0\le x\le2$. It is rotated through $2\pi$ radians about the $x$-axis to form a surface.

   (a)

   Which integral gives the area of the surface formed?

   [1 mark]

   • A$2\pi\int_0^2\sqrt{x+\frac14}\,dx$
   • B$\pi\int_0^2x\,dx$
   • C$2\pi\int_0^2\sqrt x\left(1+\frac{1}{4x}\right)dx$
   • D$2\pi\int_0^2\sqrt x\sqrt{1+\frac{1}{2\sqrt x}}\,dx$

   (b)

   Find the area of the surface.

   [1 mark]

   • A$\frac{13\pi}{2}$
   • B$\frac{9\pi}{2}$
   • C$\frac{13\pi}{3}$
   • D$\frac{13}{6}$

   (c)

   Write down an integral, in terms of $x$, for the area of the surface formed when $C$ is rotated through $2\pi$ about the $y$-axis. Do not evaluate it.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve $C$ is given parametrically by $x=3t^2$, $y=2t^3$ for $0\le t\le1$.

   (a)

   Show that the length of $C$ is given by $\int_0^1 6t\sqrt{1+t^2}\,dt$.

   [3 marks]

   (b)

   Hence find the exact length of $C$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A decorative glass bowl is modelled by rotating the curve $C$ with equation $y=\frac{x^3}{6}+\frac{1}{2x}$ for $1\le x\le2$ about the $x$-axis, where $x$ and $y$ are in centimetres.

   (a)

   Show that $1+\left(\frac{dy}{dx}\right)^2=\left(\frac{x^2}{2}+\frac{1}{2x^2}\right)^2$, and hence find the exact length of $C$.

   [6 marks]

   (b)

   Find the exact area of the glass surface of the bowl, which is the surface formed by the rotation of $C$.

   [6 marks]

   Total for question 4: 12 marks

End of questions