Area between two polar curves

Answer only one of the following two alternatives. **EITHER** Show that \(\left(n + \frac{1}{2}\right)^3 - \left(n - \frac{1}{2}\right)^3 \equiv 3n^2 + \frac{1}{4}\). [1] Use this result to prove that \(\sum_{n=1}^N n^2 = \frac{1}{6}N(N + 1)(2N + 1)\). [2] The sums \(S\), \(T\) and \(U\) are defined as follows: \begin{align} S &= 1^2 + 2^2 + 3^2 + 4^2 + \ldots + (2N)^2 + (2N + 1)^2,
T &= 1^2 + 3^2 + 5^2 + 7^2 + \ldots + (2N - 1)^2 + (2N + 1)^2,
U &= 1^2 - 2^2 + 3^2 - 4^2 + \ldots - (2N)^2 + (2N + 1)^2. \end{align} Find and simplify expressions in terms of \(N\) for each of \(S\), \(T\) and \(U\). [5] Hence

1. describe the behaviour of \(\frac{S}{T}\) as \(N \to \infty\), [1]
2. prove that if \(\frac{S}{U}\) is an integer then \(\frac{T}{U}\) is an integer. [3]

**OR** The curves \(C_1\) and \(C_2\) have polar equations $$r = 4\cos\theta \quad \text{and} \quad r = 1 + \cos\theta$$ respectively, where \(-\frac{1}{2}\pi \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Show that \(C_1\) and \(C_2\) meet at the points \(A\left(\frac{4}{3}, \alpha\right)\) and \(B\left(\frac{4}{3}, -\alpha\right)\), where \(\alpha\) is the acute angle such that \(\cos\alpha = \frac{1}{3}\). [2]
2. In a single diagram, draw sketch graphs of \(C_1\) and \(C_2\). [3]
3. Show that the area of the region bounded by the arcs \(OA\) and \(OB\) of \(C_1\), and the arc \(AB\) of \(C_2\), is $$4\pi - \frac{1}{3}\sqrt{2} - \frac{13}{2}\alpha.$$ [7]

\includegraphics{figure_1} Figure 1 Figure 1 shows the curves given by the polar equations $$r = 2, \quad 0 \leq \theta \leq \frac{\pi}{2},$$ and $$r = 1.5 + \sin 3\theta, \quad 0 \leq \theta \leq \frac{\pi}{2}.$$

1. Find the coordinates of the points where the curves intersect. [3]

The region \(S\), between the curves, for which \(r > 2\) and for which \(r < (1.5 + \sin 3\theta)\), is shown shaded in Figure 1.

1. Find, by integration, the area of the shaded region \(S\), giving your answer in the form \(a\pi + b\sqrt{3}\), where \(a\) and \(b\) are simplified fractions. [7]

The diagram above shows the curve \(C_1\) which has polar equation \(r = a(3 + 2\cos\theta)\), \(0 \leq \theta < 2\pi\) and the circle \(C_2\) with equation \(r = 4a\), \(0 \leq \theta < 2\pi\), where \(a\) is a positive constant.

1. Find, in terms of \(a\), the polar coordinates of the points where the curve \(C_1\) meets the circle \(C_2\).(4)

The regions enclosed by the curves \(C_1\) and \(C_2\) overlap and this common region \(R\) is shaded in the figure.

1. Find, in terms of \(a\), an exact expression for the area of the region \(R\).(8)
2. In a single diagram, copy the two curves in the diagram above and also sketch the curve \(C_3\) with polar equation \(r = 2a\cos\theta\), \(0 \leq \theta < 2\pi\) Show clearly the coordinates of the points of intersection of \(C_1\), \(C_2\) and \(C_3\) with the initial line, \(\theta = 0\).(3)(Total 15 marks)

\includegraphics{figure_4}

The curve \(C\) has polar equation \(r = 3a \cos \theta\), \(-\frac{\pi}{2} \leq \frac{\pi}{2}\). The curve \(D\) has polar equation \(r = a(1 + \cos \theta)\), \(-\pi \leq \theta < \pi\). Given that \(a\) is a positive constant,

1. sketch, on the same diagram, the graphs of \(C\) and \(D\), indicating where each curve cuts the initial line. [4] The graphs of \(C\) intersect at the pole \(O\) and at the points \(P\) and \(Q\).
2. Find the polar coordinates of \(P\) and \(Q\). [3]
3. Use integration to find the exact value of the area enclosed by the curve \(D\) and the lines \(\theta = 0\) and \(\theta = \frac{\pi}{3}\). [7] The region \(R\) contains all points which lie outside \(D\) and inside \(C\). Given that the value of the smaller area enclosed by the curve \(C\) and the line \(\theta = \frac{\pi}{3}\) is $$\frac{3a^2}{16}(2\pi - 3\sqrt{3}),$$
4. show that the area of \(R\) is \(\pi a^2\). [4]

\includegraphics{figure_1} A logo is designed which consists of two overlapping closed curves. The polar equations of these curves are $$r = a(3 + 2\cos \theta) \quad \text{and}$$ $$r = a(5 - 2 \cos \theta), \quad 0 \leq \theta < 2\pi.$$ Figure 1 is a sketch (not to scale) of these two curves.

1. Write down the polar coordinates of the points \(A\) and \(B\) where the curves meet the initial line. [2]
2. Find the polar coordinates of the points \(C\) and \(D\) where the two curves meet. [4]
3. Show that the area of the overlapping region, which is shaded in the figure, is $$\frac{a^2}{3}(49\pi - 48\sqrt{3}).$$ [8]

\includegraphics{figure_1} Figure 1 is a sketch of the two curves \(C_1\) and \(C_2\) with polar equations $$C_1 : r = 3a(1 - \cos \theta), \quad -\pi \leq \theta < \pi$$ and $$C_2 : r = a(1 + \cos \theta), \quad -\pi \leq \theta < \pi.$$ The curves meet at the pole \(O\), and at the points \(A\) and \(B\).

1. Find, in terms of \(a\), the polar coordinates of the points \(A\) and \(B\). [4]
2. Show that the length of the line \(AB\) is \(\frac{3\sqrt{3}}{2}a\). [2] The region inside \(C_2\) and outside \(C_1\) is shown shaded in Fig. 1.
3. Find, in terms of \(a\), the area of this region. [7] A badge is designed which has the shape of the shaded region. Given that the length of the line \(AB\) is \(4.5\) cm,
4. calculate the area of this badge, giving your answer to three significant figures. [3]

\includegraphics{figure_1} The curve \(C\) which passes through \(O\) has polar equation $$r = 4a(1 + \cos \theta), \quad -\pi < \theta \leq \pi$$ The line \(l\) has polar equation $$r = 3a \sec \theta, \quad -\frac{\pi}{2} < \theta < \frac{\pi}{2}.$$ The line \(l\) cuts \(C\) at the points \(P\) and \(Q\), as shown in Figure 1.

1. Prove that \(PQ = 6\sqrt{3}a\). [6] The region \(R\), shown shaded in Figure 1, is bounded by \(l\) and \(C\).
2. Use calculus to find the exact area of \(R\). [7]

\includegraphics{figure_8} The diagram shows two curves, \(C_1\) and \(C_2\), which intersect at the pole \(O\) and at the point \(P\). The polar equation of \(C_1\) is \(r = \sqrt{2}\cos\theta\) and the polar equation of \(C_2\) is \(r = \sqrt{2}\sin 2\theta\). For both curves, \(0 \leq \theta \leq \frac{1}{2}\pi\). The value of \(\theta\) at \(P\) is \(\alpha\).

1. Show that \(\tan\alpha = \frac{1}{2}\). [2]
2. Show that the area of the region common to \(C_1\) and \(C_2\), shaded in the diagram, is \(\frac{1}{4}\pi - \frac{1}{2}\alpha\). [7]

The diagram shows the polar curve \(C_1\) with equation \(r = 2 \sin \theta\) The diagram also shows part of the polar curve \(C_2\) with equation \(r = 1 + \cos 2\theta\) \includegraphics{figure_14}

1. On the diagram above, complete the sketch of \(C_2\) [2 marks]
2. Show that the area of the region shaded in the diagram is equal to $$k\pi + m\alpha - \sin 2\alpha + q \sin 4\alpha$$ where \(\alpha = \sin^{-1} \left( \frac{\sqrt{5} - 1}{2} \right)\), and \(k\), \(m\) and \(q\) are rational numbers. [9 marks]

The diagram shows the polar curve \(C_1\) with equation \(r = 2\sin\theta\) The diagram also shows part of the polar curve \(C_2\) with equation \(r = 1 + \cos 2\theta\) \includegraphics{figure_10}

1. On the diagram above, complete the sketch of \(C_2\) [2 marks]
2. Show that the area of the region shaded in the diagram is equal to $$k\pi + m\alpha - \sin 2\alpha + q\sin 4\alpha$$ where \(\alpha = \sin^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\), and \(k\), \(m\) and \(q\) are rational numbers. [9 marks]