4.09c Area enclosed: by polar curve

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with polar equation $$r = 1 + \tan \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The tangent to the curve \(C\) at the point \(P\) is perpendicular to the initial line.

1. Find the polar coordinates of the point \(P\). The point \(Q\) lies on the curve \(C\), where \(\theta = \frac { \pi } { 3 }\) The shaded region \(R\) is bounded by \(O P , O Q\) and the curve \(C\), as shown in Figure 1
2. Find the exact area of \(R\), giving your answer in the form $$\frac { 1 } { 2 } ( \ln p + \sqrt { q } + r )$$ where \(p , q\) and \(r\) are integers to be found.

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.

8. \begin{figure}[h] \end{figure} The curve \(C _ { 1 }\) with equation $$r = 7 \cos \theta , \quad - \frac { \pi } { 2 } < \theta \leqslant \frac { \pi } { 2 }$$ and the curve \(C _ { 2 }\) with equation $$r = 3 ( 1 + \cos \theta ) , \quad - \pi < \theta \leqslant \pi$$ are shown on Figure 1.
The curves \(C _ { 1 }\) and \(C _ { 2 }\) both pass through the pole and intersect at the point \(P\) and the point \(Q\).

1. Find the polar coordinates of \(P\) and the polar coordinates of \(Q\). The regions enclosed by the curve \(C _ { 1 }\) and the curve \(C _ { 2 }\) overlap, and the common region \(R\) is shaded in Figure 1.
2. Find the area of \(R\).

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve with polar equation $$r = 6 + a \sin \theta$$ where \(0 < a < 6\) and \(0 \leqslant \theta < 2 \pi\) The area enclosed by the curve is \(\frac { 97 \pi } { 2 }\) Find the value of the constant \(a\).

7. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 2 + \sqrt { 3 } \cos \theta , \quad 0 \leqslant \theta < 2 \pi$$ The tangent to \(C\) at the point \(P\) is parallel to the initial line.

1. Show that \(O P = \frac { 1 } { 2 } ( 3 + \sqrt { 7 } )\)
2. Find the exact area enclosed by the curve \(C\).

5. \begin{figure}[h] \end{figure} The curve \(C\), shown in Figure 1, has polar equation, \(r = 2 + \sin 3 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\) Use integration to calculate the exact value of the area enclosed by \(C\), the line \(\theta = 0\) and the line \(\theta = \frac { \pi } { 2 }\).

6. The curve \(C\) has polar equation $$r ^ { 2 } = a ^ { 2 } \cos 2 \theta , \quad \frac { - \pi } { 4 } \leq \theta \leq \frac { \pi } { 4 }$$

1. Sketch the curve \(C\).
2. Find the polar coordinates of the points where tangents to \(C\) are parallel to the initial line.
3. Find the area of the region bounded by \(C\).

8. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 1 + \sin \theta \quad - \frac { \pi } { 2 } < \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) such that the tangent to \(C\) at \(P\) is perpendicular to the initial line.

1. Use calculus to determine the polar coordinates of \(P\). The tangent to \(C\) at the point \(Q\) where \(\theta = \frac { \pi } { 2 }\) is parallel to the initial line.
   The tangent to \(C\) at \(Q\) meets the tangent to \(C\) at \(P\) at the point \(S\), as shown in Figure 1.
   The finite region \(R\), shown shaded in Figure 1, is bounded by the line segments \(Q S , S P\) and the curve \(C\).
2. Use algebraic integration to show that the area of \(R\) is $$\frac { 1 } { 32 } ( a \sqrt { 3 } + b \pi )$$ where \(a\) and \(b\) are integers to be determined.
   (6)

7. \begin{figure}[h] \end{figure} Figure 1 shows the two curves given by the polar equations $$\begin{array} { l l } r = \sqrt { 3 } \sin \theta , & 0 \leqslant \theta \leqslant \pi \\ r = 1 + \cos \theta , & 0 \leqslant \theta \leqslant \pi \end{array}$$

1. Verify that the curves intersect at the point \(P\) with polar coordinates \(\left( \frac { 3 } { 2 } , \frac { \pi } { 3 } \right)\). The region \(R\), bounded by the two curves, is shown shaded in Figure 1.
2. Use calculus to find the exact area of \(R\), giving your answer in the form \(a ( \pi - \sqrt { 3 } )\), where \(a\) is a constant to be found.
   

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5. \begin{figure}[h] \end{figure} Figure 1 shows the curves given by the polar equations $$r = 2 , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ and \(\quad r = 1.5 + \sin 3 \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\).

1. Find the coordinates of the points where the curves intersect. 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.
2. 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. $$\left[ \begin{array} { l l l } \text { Leave } \\ \text { blank } \\ \text { " } \\ \text { " } \end{array} & \\ \text { " } & \\ \text { " } & \\ \text { " } & \\ \text { " } & \\ \text { " } & \\ \text { " } & \\ \text { " } & \\ \text { " } & \end{array} \right.$$

1

1. A curve has polar equation \(r = a ( \sqrt { 2 } + 2 \cos \theta )\) for \(- \frac { 3 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\), where \(a\) is a positive constant.
   1. Sketch the curve.
   2. Find, in an exact form, the area of the region enclosed by the curve.
2. 1. Find the Maclaurin series for the function \(\mathrm { f } ( x ) = \tan \left( \frac { 1 } { 4 } \pi + x \right)\), up to the term in \(x ^ { 2 }\).
   2. Use the Maclaurin series to show that, when \(h\) is small, $$\int _ { - h } ^ { h } x ^ { 2 } \tan \left( \frac { 1 } { 4 } \pi + x \right) \mathrm { d } x \approx \frac { 2 } { 3 } h ^ { 3 } + \frac { 4 } { 5 } h ^ { 5 }$$

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 \(x ^ { 5 }\).

1

1. 1. Given that \(\mathrm { f } ( t ) = \arcsin t\), write down an expression for \(\mathrm { f } ^ { \prime } ( t )\) and show that $$\mathrm { f } ^ { \prime \prime } ( t ) = \frac { t } { \left( 1 - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$$
   2. Show that the Maclaurin expansion of the function \(\arcsin \left( x + \frac { 1 } { 2 } \right)\) begins $$\frac { \pi } { 6 } + \frac { 2 } { \sqrt { 3 } } x$$ and find the term in \(x ^ { 2 }\).
2. Sketch the curve with polar equation \(r = \frac { \pi a } { \pi + \theta }\), where \(a > 0\), for \(0 \leqslant \theta < 2 \pi\). Find, in terms of \(a\), the area of the region bounded by the part of the curve for which \(0 \leqslant \theta \leqslant \pi\) and the lines \(\theta = 0\) and \(\theta = \pi\).
3. Find the exact value of the integral $$\int _ { 0 } ^ { \frac { 3 } { 2 } } \frac { 1 } { 9 + 4 x ^ { 2 } } \mathrm {~d} x$$

3

1. Find the length of the arc of the polar curve \(r = a ( 1 + \cos \theta )\) for which \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
2. A curve \(C\) has cartesian equation \(y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x }\).
   1. The arc of \(C\) for which \(1 \leqslant x \leqslant 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a surface of revolution. Find the area of this surface. For the point on \(C\) at which \(x = 2\),
   2. show that the radius of curvature is \(\frac { 289 } { 64 }\),
   3. find the coordinates of the centre of curvature.

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 The equation of a curve, in polar coordinates, is $$r = 2 \sin 3 \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi .$$ Find the exact area of the region enclosed by the curve between \(\theta = 0\) and \(\theta = \frac { 1 } { 3 } \pi\).

6 The equation of a curve in polar coordinates is \(r = \sin 5 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 5 } \pi\).

1. Sketch the curve and write down the equations of the tangents at the pole.
2. The line of symmetry meets the curve at the pole and at one other point \(A\). Find the equation of the line of symmetry and the cartesian coordinates of \(A\).
3. Find the area of the region enclosed by this curve.

4 The equation of a curve, in polar coordinates, is $$r = 2 \cos 2 \theta \quad ( - \pi < \theta \leqslant \pi ) .$$

1. Find the values of \(\theta\) which give the directions of the tangents at the pole. One loop of the curve is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-3_362_720_653_708}
2. Find the exact value of the area of the region enclosed by the loop.

1

1. A curve has polar equation \(r = a \cos 3 \theta\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), where \(a\) is a positive constant.
   1. Sketch the curve, using a continuous line for sections where \(r > 0\) and a broken line for sections where \(r < 0\).
   2. Find the area enclosed by one of the loops.
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x\).
3. Use a trigonometric substitution to find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + 3 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\).

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, \(\int _ { h } ^ { 4 h } \frac { \tan x } { x } \mathrm {~d} x \approx 3 h + 7 h ^ { 3 }\).

1

1. Fig. 1 shows the curve with polar equation \(r = a ( 1 - \cos 2 \theta )\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{43b4c7ed-3556-4d87-8aef-0111fe747982-2_529_620_577_799} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Find the area of the region enclosed by the curve.
2. 1. Given that \(\mathrm { f } ( x ) = \arctan ( \sqrt { 3 } + x )\), find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
   2. Hence find the Maclaurin series for \(\arctan ( \sqrt { 3 } + x )\), as far as the term in \(x ^ { 2 }\).
   3. Hence show that, if \(h\) is small, \(\int _ { - h } ^ { h } x \arctan ( \sqrt { 3 } + x ) \mathrm { d } x \approx \frac { 1 } { 6 } h ^ { 3 }\).

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curves with polar equations $$\begin{array} { l l } r = 2 \sin \theta & 0 \leqslant \theta \leqslant \pi \\ r = 1.5 - \sin \theta & 0 \leqslant \theta \leqslant 2 \pi \end{array}$$ The curves intersect at the points \(P\) and \(Q\).

1. Find the polar coordinates of the point \(P\) and the polar coordinates of the point \(Q\). The region \(R\), shown shaded in Figure 1, is enclosed by the two curves.
2. Find the exact area of \(R\), giving your answer in the form \(p \pi + q \sqrt { 3 }\), where \(p\) and \(q\) are rational numbers to be found.

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)