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

1. 1
   A spiral has polar equation $r=2\theta$ for $0\le\theta\le\pi$.

   (a)

   Which integral gives the area enclosed by the spiral and the half-line $\theta=\pi$?

   [1 mark]

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

   (b)

   Find the area enclosed by the spiral and the half-line $\theta=\pi$.

   [1 mark]

   • A$\frac{4\pi^3}{3}$
   • B$\frac{\pi^3}{3}$
   • C$\frac{\pi^2}{2}$
   • D$\frac{2\pi^3}{3}$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   Which expression is equal to $r^2$?

   [1 mark]

   • A$9+4\cos^2\theta$
   • B$9+6\cos\theta+4\cos^2\theta$
   • C$9+12\cos\theta+4\cos^2\theta$
   • D$3+4\cos\theta+2\cos^2\theta$

   (b)

   Find the value of $\int_0^{2\pi}\cos^2\theta\,d\theta$.

   [1 mark]

   • A$\pi$
   • B$2\pi$
   • C$0$
   • D$\frac{\pi}{2}$

   (c)

   Hence find the area enclosed by $C$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve $C$ has polar equation $r=4\sin2\theta$ for $0\le\theta\le\frac{\pi}{2}$, forming one loop from the pole.

   (a)

   Show that the area enclosed by the loop is $2\pi$.

   [3 marks]

   (b)

   Find the exact area of the region bounded by $C$ and the half-lines $\theta=0$ and $\theta=\frac{\pi}{6}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The cardioid $C_1$ has polar equation $r=1+\cos\theta$ and the circle $C_2$ has polar equation $r=3\cos\theta$, where $-\pi\le\theta\le\pi$ for $C_1$ and $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$ for $C_2$.

   (a)

   (i) Show that the area enclosed by $C_1$ is $\frac{3\pi}{2}$.
   (ii) Use integration to find the area enclosed by $C_2$.

   [6 marks]

   (b)

   Find the polar coordinates of the points where $C_1$ and $C_2$ meet, and hence show that the area inside $C_2$ but outside $C_1$ is $\pi$.

   [6 marks]

   Total for question 4: 12 marks

End of questions