Area enclosed by a polar curveEdexcel A-Level Further Maths: Subtopic test

1. 1
   The curve $C$ has polar equation $r=2\theta$ for $0\le\theta\le\pi$.

   (a)

   Which integral gives the area of the region bounded by $C$ and the half-lines $\theta=0$ and $\theta=\frac{\pi}{2}$?

   [1 mark]

   • A$\frac12\int_0^{\pi/2}4\theta^2\,d\theta$
   • B$\frac12\int_0^{\pi/2}2\theta\,d\theta$
   • C$\int_0^{\pi/2}4\theta^2\,d\theta$
   • D$\frac12\int_0^{\pi/2}2\theta^2\,d\theta$

   (b)

   Find the area of the region described in part (a).

   [1 mark]

   • A$\frac{\pi^3}{6}$
   • B$\frac{\pi^2}{8}$
   • C$\frac{\pi^3}{24}$
   • D$\frac{\pi^3}{12}$

   (c)

   Find the exact area of the region bounded by $C$ and the half-lines $\theta=\frac{\pi}{2}$ and $\theta=\pi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has polar equation $r=3+2\cos\theta$ for $0\le\theta\le2\pi$.

   (a)

   Which expression equals $r^2$ on $C$?

   [1 mark]

   • A$9+4\cos^2\theta$
   • B$11+12\cos\theta+2\cos2\theta$
   • C$13+12\cos\theta+2\cos2\theta$
   • D$9+12\cos\theta+4\cos2\theta$

   (b)

   Find the area enclosed by $C$.

   [1 mark]

   • A$\frac{11\pi}{2}$
   • B$22\pi$
   • C$11\pi$
   • D$13\pi$

   (c)

   Find the exact area of the region bounded by $C$ and the half-lines $\theta=0$ and $\theta=\frac{\pi}{2}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has polar equation $r=2(1+\cos\theta)$ for $0\le\theta\le\pi$.

   (a)

   Find the polar coordinates of the point of $C$, other than the pole and the point where $\theta=0$, at which the tangent is perpendicular to the initial line.

   [3 marks]

   (b)

   Find the greatest distance of a point of $C$ from the initial line, giving your answer in exact form.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C_1$ has polar equation $r=4\cos\theta$ and the curve $C_2$ has polar equation $r=2$, for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$.

   (a)

   (i) Show that $C_1$ and $C_2$ meet where $\theta=\pm\frac{\pi}{3}$.
   (ii) Show that the area of the region inside $C_1$ and outside $C_2$ is $\int_0^{\pi/3}\left(16\cos^2\theta-4\right)d\theta$.

   [6 marks]

   (b)

   (i) Hence find the exact area of the region inside $C_1$ and outside $C_2$.
   (ii) $C_1$ is a circle of radius $2$. Find the exact area of the region inside both $C_1$ and $C_2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions