Area of region with line boundary

The curve \(C\) has polar equation \(r = a ( 1 - \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\).

1. Sketch \(C\).
2. Find the area of the region enclosed by the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\), the half-line \(\theta = \frac { 1 } { 2 } \pi\) and the half-line \(\theta = \frac { 3 } { 2 } \pi\).
3. Show that $$\left( \frac { \mathrm { d } s } { \mathrm {~d} \theta } \right) ^ { 2 } = 4 a ^ { 2 } \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\).

10 The curve \(C\) has polar equation \(r = 3 + 2 \cos \theta\), for \(- \pi < \theta \leqslant \pi\). The straight line \(l\) has polar equation \(r \cos \theta = 2\). Sketch both \(C\) and \(l\) on a single diagram. Find the polar coordinates of the points of intersection of \(C\) and \(l\). The region \(R\) is enclosed by \(C\) and \(l\), and contains the pole. Find the area of \(R\).

5 The curve \(C\) has polar equation \(r = 1 + 2 \cos \theta\). Sketch the curve for \(- \frac { 2 } { 3 } \pi \leqslant \theta < \frac { 2 } { 3 } \pi\). Find the area bounded by \(C\) and the half-lines \(\theta = - \frac { 1 } { 3 } \pi , \theta = \frac { 1 } { 3 } \pi\).

1 The curve \(C\) has polar equation \(r = 2 \mathrm { e } ^ { \theta }\), for \(\frac { 1 } { 6 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Find

1. the area of the region bounded by the half-lines \(\theta = \frac { 1 } { 6 } \pi , \theta = \frac { 1 } { 2 } \pi\) and \(C\),
2. the length of \(C\).

2 The diagram shows a sketch of part of the curve \(C\) whose polar equation is \(r = 1 + \tan \theta\). The point \(O\) is the pole. \includegraphics[max width=\textwidth, alt={}, center]{0c177d90-02ae-4e91-bc9d-d0c7051799b8-3_561_629_406_772} The points \(P\) and \(Q\) on the curve are given by \(\theta = 0\) and \(\theta = \frac { \pi } { 3 }\) respectively.

1. Show that the area of the region bounded by the curve \(C\) and the lines \(O P\) and \(O Q\) is $$\frac { 1 } { 2 } \sqrt { 3 } + \ln 2$$ (6 marks)
2. Hence find the area of the shaded region bounded by the line \(P Q\) and the arc \(P Q\) of \(C\).

8 The diagram shows a sketch of a curve \(C\) and a line \(L\), which is parallel to the initial line and touches the curve at the points \(P\) and \(Q\). \includegraphics[max width=\textwidth, alt={}, center]{32de7ef6-b7aa-4bfd-a73a-e12bfc0147e2-5_506_762_447_639} The polar equation of the curve \(C\) is $$r = 4 ( 1 - \sin \theta ) , \quad 0 \leqslant \theta < 2 \pi$$ and the polar equation of the line \(L\) is $$r \sin \theta = 1$$

1. Show that the polar coordinates of \(P\) are \(\left( 2 , \frac { \pi } { 6 } \right)\) and find the polar coordinates of \(Q\).
2. Find the area of the shaded region \(R\) bounded by the line \(L\) and the curve \(C\). Give your answer in the form \(m \sqrt { 3 } + n \pi\), where \(m\) and \(n\) are integers.

8 The polar equation of a curve \(C\) is $$r = 5 + 2 \cos \theta , \quad - \pi \leqslant \theta \leqslant \pi$$

1. Verify that the points \(A\) and \(B\), with polar coordinates ( 7,0 ) and ( \(3 , \pi\) ) respectively, lie on the curve \(C\).
2. Sketch the curve \(C\).
3. Find the area of the region bounded by the curve \(C\).
4. The point \(P\) is the point on the curve \(C\) for which \(\theta = \alpha\), where \(0 < \alpha \leqslant \frac { \pi } { 2 }\). The point \(Q\) lies on the curve such that \(P O Q\) is a straight line, where the point \(O\) is the pole. Find, in terms of \(\alpha\), the area of triangle \(O Q B\).

8 The diagram shows the sketch of part of a curve, the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-20_609_670_358_703} The polar equation of the curve is \(r = 1 + \tan \theta\).
The point \(A\) is the point on the curve at which \(\theta = \frac { \pi } { 3 }\).
The perpendicular, \(A N\), from \(A\) to the initial line intersects the curve at the point \(B\).

1. Find the exact length of \(O A\).
2. 1. Given that, at the point \(B , \theta = \alpha\), show that \(( \cos \alpha + \sin \alpha ) ^ { 2 } = 1 + \frac { \sqrt { 3 } } { 2 }\).
   2. Hence, or otherwise, find \(\alpha\) in terms of \(\pi\).
3. Show that the area of triangle \(O A B\) is \(\frac { 3 + 2 \sqrt { 3 } } { 6 }\).
4. Find, in an exact simplified form, the area of the shaded region bounded by the curve and the line segment \(A B\).
   [0pt] [7 marks]
   \includegraphics[max width=\textwidth, alt={}]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-23_2488_1709_219_153}
   \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

4. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 4 + \cos 2 \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the value of \(r\) is \(\frac { 9 } { 2 }\) The point \(N\) lies on the initial line and \(A N\) is perpendicular to the initial line.
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the initial line and the line \(A N\). Find the exact area of the shaded region \(R\), giving your answer in the form \(p \pi + q \sqrt { 3 }\) where \(p\) and \(q\) are rational numbers to be found.

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = 2 \sqrt { \sinh \theta + \cosh \theta } \quad 0 \leqslant \theta \leqslant \pi$$ The region \(R\), shown shaded in Figure 1, is bounded by the initial line, the curve and the line with equation \(\theta = \pi\) Use algebraic integration to determine the exact area of \(R\) giving your answer in the form \(p \mathrm { e } ^ { q } - r\) where \(p , q\) and \(r\) are real numbers to be found.

9

1. The line \(L\) has polar equation $$r = \frac { 7 } { 4 } \sec \theta \quad \left( - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 } \right)$$ Show that \(L\) is perpendicular to the initial line.
   9
2. The curve \(C\) has polar equation $$r = 3 + \cos \theta \quad ( - \pi < \theta \leq \pi )$$ Find the polar coordinates of the points of intersection of \(L\) and \(C\) Fully justify your answer.
   9
3. The region \(R\) is the set of points such that
   and $$r > \frac { 7 } { 4 } \sec \theta$$ Find the exact area of \(R\) $$r < 3 + \cos \theta$$ Find the exact area of \(R\) [0pt] [7 marks]

The curve \(C\) has polar equation \(r = a ( 1 - \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\). Sketch \(C\). Find the area of the region enclosed by the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\), the half-line \(\theta = \frac { 1 } { 2 } \pi\) and the half-line \(\theta = \frac { 3 } { 2 } \pi\). Show that $$\left( \frac { \mathrm { d } s } { \mathrm {~d} \theta } \right) ^ { 2 } = 4 a ^ { 2 } \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) ,$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\). {www.cie.org.uk} after the live examination series.
}

11 A curve has polar equation \(r = \mathrm { e } ^ { \sin \theta }\) for \(- \pi < \theta \leqslant \pi\).

1. State the polar coordinates of the point where the curve crosses the initial line.
2. State also the polar coordinates of the points where \(r\) takes its least and greatest values.
3. Sketch the curve.
4. By deriving a suitable Maclaurin series up to and including the term in \(\theta ^ { 2 }\), find an approximation, to 3 decimal places, for the area of the region enclosed by the curve, the initial line and the line \(\theta = 0.3\).

3

1. Sketch the curve with polar equation \(r = \frac { 1 } { 1 + \theta } , 0 \leqslant \theta \leqslant 2 \pi\).
2. Find, in terms of \(\pi\), the area of the region enclosed by the curve and the part of the initial line between the endpoints of the curve.

8 The curve \(C\) has polar equation \(r = \theta ^ { 2 } + 2 \theta\) for \(0 \leq \theta \leq 3\).

1. Find the area of the region enclosed by \(C\) and the half-lines \(\theta = 0\) and \(\theta = 3\).
2. Determine the length of \(C\).

4

1. Draw a sketch of the curve \(C\) whose polar equation is \(r = \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
2. On the same diagram draw the line \(\theta = \alpha\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\). The region bounded by \(C\) and the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(R\).
3. Find the exact value of \(\alpha\) for which the line \(\theta = \alpha\) divides \(R\) into two regions of equal area.

Answer only one of the following two alternatives. EITHER The points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{i}\), \(2\mathbf{j}\) and \(4\mathbf{k}\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(ABC\). The point \(P\) on the line-segment \(ON\) is such that \(OP = \frac{3}{4}ON\). The line \(AP\) meets the plane \(OBC\) at \(Q\). Find a vector perpendicular to the plane \(ABC\) and show that the length of \(ON\) is \(\frac{1}{\sqrt{(21)}}\). [4] Find the position vector of the point \(Q\). [5] Show that the acute angle between the planes \(ABC\) and \(ABQ\) is \(\cos^{-1}\left(\frac{4}{5}\right)\). [5] OR The curve \(C\) has polar equation \(r = a(1 - \cos\theta)\) for \(0 \leqslant \theta < 2\pi\). Sketch \(C\). [2] Find the area of the region enclosed by the arc of \(C\) for which \(\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\), the half-line \(\theta = \frac{1}{3}\pi\) and the half-line \(\theta = \frac{2}{3}\pi\). [5] Show that $$\left(\frac{\mathrm{d}s}{\mathrm{d}\theta}\right)^2 = 4a^2\sin^2\left(\frac{1}{2}\theta\right),$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\). [7]

The curve \(C\) has polar equation $$r = 5\sqrt{\cot \theta},$$ where \(0.01 \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Find the area of the finite region bounded by \(C\) and the line \(\theta = 0.01\), showing full working. Give your answer correct to \(1\) decimal place. [3]

Let \(P\) be the point on \(C\) where \(\theta = 0.01\).

1. Find the distance of \(P\) from the initial line, giving your answer correct to \(1\) decimal place. [2]
2. Find the maximum distance of \(C\) from the initial line. [3]
3. Sketch \(C\). [2]

\includegraphics{figure_7} The diagram shows the curve with equation, in polar coordinates, $$r = 3 + 2\cos \theta, \quad \text{for } 0 \leq \theta < 2\pi.$$ The points \(P\), \(Q\), \(R\) and \(S\) on the curve are such that the straight lines \(POR\) and \(QOS\) are perpendicular, where \(O\) is the pole. The point \(P\) has polar coordinates \((r, \alpha)\).

1. Show that \(OP + OQ + OR + OS = k\), where \(k\) is a constant to be found. [3]
2. Given that \(\alpha = \frac{1}{4}\pi\), find the exact area bounded by the curve and the lines \(OP\) and \(OQ\) (shaded in the diagram). [5]

The curve \(C\) has polar equation \(r = 2 + \tan \theta\) The curve \(C\) meets the line \(\theta = \frac{\pi}{4}\) at the point \(A\) The point \(B\) has polar coordinates \((4, 0)\) The diagram shows part of the curve \(C\), and the points \(A\) and \(B\) \includegraphics{figure_16}

1. Show that the area of triangle \(OAB\) is \(3\sqrt{2}\) units. [2 marks]
2. Find the area of the shaded region. Give your answer in an exact form. [7 marks]

The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. \includegraphics{figure_11} The polar equation of \(C\) is \(r = 4 + 2\cos \theta\), \quad \(-\pi \leq \theta \leq \pi\)

1. Show that the area of the region bounded by the curve \(C\) is \(18\pi\) [4 marks]
2. Points \(A\) and \(B\) lie on the curve \(C\) such that \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\) and \(AOB\) is an equilateral triangle. Find the polar equation of the line segment \(AB\) [4 marks]

The curve \(C\) has polar equation \(r = 3(2 + \cos \theta)\), \(0 \leq \theta \leq \pi\). Determine the area enclosed between \(C\) and the initial line. Give your answer in the form \(\frac{a}{b}\pi\), where \(a\) and \(b\) are positive integers whose values are to be found. [5]