Area in polar coordinatesEdexcel International A Level Further Maths: Subtopic test

1. 1
   The curve $C$ has polar equation $r=4\cos\theta$, for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$.

   (a)

   Which expression gives the area enclosed by $C$?

   [1 mark]

   • A$\frac12\int_{-\pi/2}^{\pi/2}4\cos\theta\,d\theta$
   • B$\frac12\int_{-\pi/2}^{\pi/2}16\cos^2\theta\,d\theta$
   • C$\int_{-\pi/2}^{\pi/2}16\cos^2\theta\,d\theta$
   • D$\frac12\int_{0}^{\pi/2}16\cos^2\theta\,d\theta$

   (b)

   Find the area enclosed by $C$.

   [1 mark]

   • A$2\pi$
   • B$8\pi$
   • C$4\pi$
   • D$16\pi$

   (c)

   Find the exact area of the region bounded by $C$ and the half-lines $\theta=0$ and $\theta=\frac{\pi}{4}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The spiral $S$ has polar equation $r=2\theta$, for $\theta\ge0$.

   (a)

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

   [1 mark]

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

   (b)

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

   [1 mark]

   • A$\frac{7\pi^3}{6}$
   • B$\frac{\pi^3}{12}$
   • C$\frac{2\pi^3}{3}$
   • D$\frac{7\pi^3}{12}$

   (c)

   Find the exact area of the region bounded by $S$ and the half-lines $\theta=\pi$ and $\theta=2\pi$.

   [2 marks]

   Total for question 2: 4 marks

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

   (a)

   Show that the area enclosed by $C$ is $11\pi$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C_1$ has polar equation $r=3\cos\theta$, for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$, and the curve $C_2$ has polar equation $r=1+\cos\theta$, for $-\pi\le\theta\le\pi$. The curves meet at the pole and at two other points.

   (a)

   Show that the curves meet where $\theta=\pm\frac{\pi}{3}$, and find the exact area of the region that lies inside both $C_1$ and $C_2$.

   [6 marks]

   (b)

   Hence find the exact area of the region that lies inside $C_2$ but outside $C_1$.

   [6 marks]

   Total for question 4: 12 marks

End of questions