Area of region with line boundary

5 The curve \(C\) has polar equation \(r = \operatorname { acot } \left( \frac { 1 } { 3 } \pi - \theta \right)\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 6 } \pi\). It is given that the greatest distance of a point on \(C\) from the pole is \(2 \sqrt { 3 }\).

1. Sketch \(C\) and show that \(a = 2\).
2. Find the exact value of the area of the region bounded by \(C\), the initial line and the half-line \(\theta = \frac { 1 } { 6 } \pi\).
3. Show that \(C\) has Cartesian equation \(2 ( x + y \sqrt { 3 } ) = ( x \sqrt { 3 } - y ) \sqrt { x ^ { 2 } + y ^ { 2 } }\).

5 The curve \(C\) has polar equation \(r = \frac { 1 } { \pi - \theta } - \frac { 1 } { \pi }\), where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Sketch \(C\).
2. Show that the area of the region bounded by the half-line \(\theta = \frac { 1 } { 2 } \pi\) and \(C\) is \(\frac { 3 - 4 \ln 2 } { 4 \pi }\).

5 The curve \(C\) has polar equation \(r = 3 + 2 \sin \theta\), for \(- \pi < \theta \leqslant \pi\).

1. The diagram shows part of \(C\). Sketch the rest of \(C\) on the diagram.
   
   [diagram]
   The straight line \(l\) has polar equation \(r \sin \theta = 2\).
2. Add \(l\) to the diagram in part (a) and find the polar coordinates of the points of intersection of \(C\) and \(l\).
3. The region \(R\) is enclosed by \(C\) and \(l\), and contains the pole. Find the area of \(R\), giving your answer in exact form.

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = 10 \cos \theta + \tan \theta \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P\) lies on the curve where \(\theta = \frac { \pi } { 3 }\) The region \(R\), shown shaded in Figure 1, is bounded by the initial line, the curve and the line \(O P\), where \(O\) is the pole. Use algebraic integration to show that the exact area of \(R\) is $$\frac { 1 } { 12 } ( a \pi + b \sqrt { 3 } + c )$$ where \(a\), \(b\) and \(c\) are integers to be determined.

10. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with polar equation $$r = 1 + \cos \theta \quad 0 \leqslant \theta \leqslant \pi$$ and the line \(l\) with polar equation $$r = k \sec \theta \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ where \(k\) is a positive constant.
Given that

• \(\quad C\) and \(l\) intersect at the point \(P\)
• \(O P = 1 + \frac { \sqrt { 3 } } { 2 }\)
  1. determine the exact value of \(k\).

The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\), the initial line and \(l\).

Use algebraic integration to show that the area of \(R\) is $$p \pi + q \sqrt { 3 } + r$$ where \(p , q\) and \(r\) are simplified rational numbers to be determined.

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

1. Prove that \(P Q = 6 \sqrt { } 3 a\). The region \(R\), shown shaded in the diagram, is bounded by \(l\) and \(C\).
2. Use calculus to find the exact area of \(R\). \includegraphics[max width=\textwidth, alt={}, center]{d9aa1f75-ef35-4bf0-85c2-dff36872ca46-2_714_778_1959_1153}

4. \includegraphics[max width=\textwidth, alt={}, center]{d6befd60-de40-41b6-8ae5-48656dbca40c-3_535_1027_276_577} The diagram above shows a sketch of the curve \(C\) with polar equation $$r = 4 \sin \theta \cos ^ { 2 } \theta , \quad 0 \leq \theta < \frac { \pi } { 2 }$$ The tangent to \(C\) at the point \(P\) is perpendicular to the initial line.

1. Show that \(P\) has polar coordinates \(\left( \frac { 3 } { 2 } , \frac { \pi } { 6 } \right)\). The point \(Q\) on \(C\) has polar coordinates \(\left( \sqrt { 2 } , \frac { \pi } { 4 } \right)\).
   The shaded region \(R\) is bounded by \(O P , O Q\) and \(C\), as shown in the diagram above.
2. Show that the area of \(R\) is given by $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \sin ^ { 2 } 2 \theta \cos 2 \theta + \frac { 1 } { 2 } - \frac { 1 } { 2 } \cos 4 \theta \right) \mathrm { d } \theta$$
3. Hence, or otherwise, find the area of \(R\), giving your answer in the form \(a + b \pi\), where \(a\) and \(b\) are rational numbers.
   (Total 14 marks)

6. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 2 + \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the value of \(r\) is \(\frac { 5 } { 2 }\).
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 \(R\).

6. \begin{figure}[h] \end{figure} The curve \(C\), shown in Figure 1, has polar equation $$r = 3 a ( 1 + \cos \theta ) , \quad 0 \leqslant \theta < \pi$$ The tangent to \(C\) at the point \(A\) is parallel to the initial line.

1. Find the polar coordinates of \(A\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the initial line and the line \(O A\).
2. Use calculus to find the area of the shaded region \(R\), giving your answer in the form \(a ^ { 2 } ( p \pi + q \sqrt { 3 } )\), where \(p\) and \(q\) are rational numbers.

1

1. A curve has polar equation \(r = a ( 1 - \cos \theta )\), where \(a\) is a positive constant.
   1. Sketch the curve.
   2. Find the area of the region enclosed by the section of the curve for which \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\) and the line \(\theta = \frac { 1 } { 2 } \pi\).
2. Use a trigonometric substitution to show that \(\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { 1 } { 4 \sqrt { 3 } }\).
3. In this part of the question, \(\mathrm { f } ( x ) = \arccos ( 2 x )\).
   1. Find \(\mathrm { f } ^ { \prime } ( x )\).
   2. Use a standard series to expand \(\mathrm { f } ^ { \prime } ( x )\), and hence find the series for \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to the term in \(x ^ { 5 }\).

9 The equation of a curve, in polar coordinates, is $$r = \sec \theta + \tan \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi$$

1. Sketch the curve.
2. Find the exact area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 3 } \pi\).
3. Find a cartesian equation of the curve.

4 The equation of a curve, in polar coordinates, is $$r = 1 + 2 \sec \theta , \quad \text { for } - \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi$$

1. Find the exact area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 6 } \pi\). [The result \(\int \sec \theta \mathrm { d } \theta = \ln | \sec \theta + \tan \theta |\) may be assumed.]
2. Show that a cartesian equation of the curve is \(( x - 2 ) \sqrt { x ^ { 2 } + y ^ { 2 } } = x\).

7 The equation of a curve, in polar coordinates, is $$r = \sqrt { 3 } + \tan \theta , \quad \text { for } - \frac { 1 } { 3 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$$

1. Find the equation of the tangent at the pole.
2. State the greatest value of \(r\) and the corresponding value of \(\theta\).
3. Sketch the curve.
4. Find the exact area of the region enclosed by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 4 } \pi\).

1

1. A curve has polar equation \(r = a \mathrm { e } ^ { - k \theta }\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) and \(k\) are positive constants. The points A and B on the curve correspond to \(\theta = 0\) and \(\theta = \pi\) respectively.
   1. Sketch the curve.
   2. Find the area of the region enclosed by the curve and the line AB .
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { 3 + 4 x ^ { 2 } } \mathrm {~d} x\).
3. 1. Find the Maclaurin series for \(\tan x\), up to the term in \(x ^ { 3 }\).
   2. Use this Maclaurin series to show that, when \(h\) is small, \(\int _ { h } ^ { 4 h } \frac { \tan x } { x } \mathrm {~d} x \approx 3 h + 7 h ^ { 3 }\).

3

1. A curve has polar equation \(r = a \tan \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\), where \(a\) is a positive constant.
   1. Sketch the curve.
   2. Find the area of the region between the curve and the line \(\theta = \frac { 1 } { 4 } \pi\). Indicate this region on your sketch.
2. 1. Find the eigenvalues and corresponding eigenvectors for the matrix \(\mathbf { M }\) where $$\mathbf { M } = \left( \begin{array} { l l } 0.2 & 0.8 \\ 0.3 & 0.7 \end{array} \right)$$
   2. Give a matrix \(\mathbf { Q }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } = \mathbf { Q D } \mathbf { Q } ^ { - 1 }\). Section B (18 marks)

5 Draw a sketch of the curve \(C\) whose polar equation is \(r = \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). 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 \(R\). Find the exact value of \(\alpha\) for which the line \(\theta = \alpha\) divides \(R\) into two regions of equal area.

2 The curve \(C\) has polar equation $$r = a \left( 1 - \mathrm { e } ^ { - \theta } \right)$$ where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\).

1. Draw a sketch of \(C\).
2. Show that the area of the region bounded by \(C\) and the lines \(\theta = \ln 2\) and \(\theta = \ln 4\) is $$\frac { 1 } { 2 } a ^ { 2 } \left( \ln 2 - \frac { 13 } { 32 } \right)$$

4 The curve \(C\) has polar equation \(r = 2 + 2 \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\). Sketch the graph of \(C\). Find the area of the region \(R\) enclosed by \(C\) and the initial line. The half-line \(\theta = \frac { 1 } { 5 } \pi\) divides \(R\) into two parts. Find the area of each part, correct to 3 decimal places.

5 The curve \(C\) has polar equation \(r = a ( 1 + \sin \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\). Draw a sketch of \(C\). Find the exact value of the area of the region enclosed by \(C\) and the half-lines \(\theta = \frac { 1 } { 3 } \pi\) and \(\theta = \frac { 2 } { 3 } \pi\).

4 A curve \(C\) has polar equation \(r ^ { 2 } = 8 \operatorname { cosec } 2 \theta\) for \(0 < \theta < \frac { 1 } { 2 } \pi\). Find a cartesian equation of \(C\). Sketch \(C\). Determine the exact area of the sector bounded by the arc of \(C\) between \(\theta = \frac { 1 } { 6 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\), the half-line \(\theta = \frac { 1 } { 6 } \pi\) and the half-line \(\theta = \frac { 1 } { 3 } \pi\).
[0pt] [It is given that \(\int \operatorname { cosec } x \mathrm {~d} x = \ln \left| \tan \frac { 1 } { 2 } x \right| + c\).]

2 The curve \(C\) has polar equation \(r ^ { 2 } = \ln ( 1 + \theta )\), for \(0 \leqslant \theta \leqslant 2 \pi\).

1. Sketch \(C\).
2. Using the substitution \(u = 1 + \theta\), or otherwise, find the area of the region bounded by \(C\) and the initial line, leaving your answer in an exact form.

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

1. Use the fact that \(\frac { \sin \theta } { \theta }\) tends to 1 as \(\theta\) tends to 0 to show that the line with carresian equation \(y = 1\) is an asymptote to \(C\).
2. Sketch \(C\). The points \(P\) and \(Q\) on \(C\) correspond to \(\theta = \frac { 1 } { 6 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\) respectively.
3. Find the area of the sector \(O P Q\), where \(O\) is the origin.
4. Show that the length of the are \(P Q\) is $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \sqrt { } \left( 1 + \theta ^ { 2 } \right) } { \theta ^ { 2 } } \mathrm {~d} \theta$$

7 The curve \(C\) has equation $$r = 10 \ln ( 1 + \theta )$$ where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Draw a sketch of \(C\). Use the substitution \(w = \ln ( 1 + \theta )\) to show that the area of the sector bounded by the line \(\theta = \frac { 1 } { 2 } \pi\) and the arc of \(C\) joining the origin to the point where \(\theta = \frac { 1 } { 2 } \pi\) is $$50 \left( b ^ { 2 } - 2 b + 2 \right) \mathrm { e } ^ { b } - 100$$ where \(b = \ln \left( 1 + \frac { 1 } { 2 } \pi \right)\).

3 The curve \(C\) has polar equation $$r = \left( \frac { 1 } { 2 } \pi - \theta \right) ^ { 2 } ,$$ where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Draw a sketch of \(C\). Find the area of the region bounded by \(C\) and the initial line, leaving your answer in terms of \(\pi\).

8 The curve \(C\) has polar equation \(r = 1 + \sin \theta\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Draw a sketch of \(C\). The area of the region enclosed by the initial line, the half-line \(\theta = \frac { 1 } { 2 } \pi\), and the part of \(C\) for which \(\theta\) is positive, is denoted by \(A _ { 1 }\). The area of the region enclosed by the initial line, and the part of \(C\) for which \(\theta\) is negative, is denoted by \(A _ { 2 }\). Find the ratio \(A _ { 1 } : A _ { 2 }\), giving your answer correct to 1 decimal place.