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

1. 1
   The curve $C$ has equation $y=\frac23x^{\frac32}$ for $x\geq0$.

   (a)

   Find $\frac{dy}{dx}$.

   [1 mark]

   • A$\frac23x^{\frac12}$
   • B$x^{\frac32}$
   • C$x^{\frac12}$
   • D$\frac49x^{-\frac12}$

   (b)

   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$\int_0^3\sqrt{1+x}\,dx$
   • C$\int_0^3\sqrt{1+x^{\frac32}}\,dx$
   • D$\int_0^3\sqrt{1+\sqrt{x}}\,dx$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve is given parametrically by $x=3t^2$, $y=2t^3$ for $0\leq t\leq1$.

   (a)

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

   [1 mark]

   • A$36t^2+36t^4$
   • B$36t^2+72t^3+36t^4$
   • C$6t+6t^2$
   • D$6t\sqrt{1+t^2}$

   (b)

   Which integral gives the length of the curve?

   [1 mark]

   • A$\int_0^1 36t^2(1+t^2)\,dt$
   • B$\int_0^1(6t+6t^2)\,dt$
   • C$\int_0^1\sqrt{1+t^2}\,dt$
   • D$\int_0^1 6t\sqrt{1+t^2}\,dt$

   (c)

   Find the exact length of the curve.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=\sqrt{x}$ for $0\leq x\leq2$. The arc of $C$ is rotated through $2\pi$ radians about the $x$-axis to form a curved surface of area $S$.

   (a)

   Show that $S=2\pi\int_0^2\sqrt{x+\frac14}\,dx$.

   [3 marks]

   (b)

   Hence find the exact value of $S$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A cardioid $C$ has polar equation $r=2(1+\cos\theta)$ for $0\leq\theta\leq\pi$. A formula booklet may be used.

   (a)

   Show that the length of $C$ is $8$.

   [6 marks]

   (b)

   The arc $C$ is rotated through $2\pi$ radians about the initial line. Using the result $ds=4\cos\frac{\theta}{2}\,d\theta$ for a small element of arc, find the exact area of the surface formed.

   [6 marks]

   Total for question 4: 12 marks

End of questions