4.09c Area enclosed: by polar curve

9. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C _ { 1 }\) with polar equation \(r = 2 a \sin 2 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\), and the circle \(C _ { 2 }\) with polar equation \(r = a , 0 \leqslant \theta \leqslant 2 \pi\), 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 \(C _ { 2 }\) The regions enclosed by the curve \(C _ { 1 }\) and the circle \(C _ { 2 }\) overlap and the common region \(R\) is shaded in Figure 1.
2. Find the area of the shaded region \(R\), giving your answer in the form \(\frac { 1 } { 12 } a ^ { 2 } ( p \pi + q \sqrt { 3 } )\), where \(p\) and \(q\) are integers to be found.

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.

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with polar equations $$\begin{array} { l l } C _ { 1 } : r = \frac { 3 } { 2 } \cos \theta , & 0 \leqslant \theta \leqslant \frac { \pi } { 2 } \\ C _ { 2 } : r = 3 \sqrt { 3 } - \frac { 9 } { 2 } \cos \theta , & 0 \leqslant \theta \leqslant \frac { \pi } { 2 } \end{array}$$ The curves intersect at the point \(P\).

1. Find the polar coordinates of \(P\). The region \(R\), shown shaded in Figure 1, is enclosed by the curves \(C _ { 1 }\) and \(C _ { 2 }\) and the initial line.
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.

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with polar equation $$r = 4 \cos 2 \theta , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 } \text { and } \frac { 3 \pi } { 4 } \leqslant \theta \leqslant \frac { 5 \pi } { 4 }$$ The lines \(P Q , Q R , R S\) and \(S P\) are tangents to \(C\), where \(Q R\) and \(S P\) are parallel to the initial line and \(P Q\) and \(R S\) are perpendicular to the initial line.

1. Find the polar coordinates of the points where the tangent SP touches the curve. Give the values of \(\theta\) to 3 significant figures.
2. Find the exact area of the finite region bounded by the curve \(C\), shown unshaded in Figure 1.
3. Find the area enclosed by the rectangle \(P Q R S\) but outside the curve \(C\), shown shaded in Figure 1.

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

1. Determine the polar coordinates of \(A\). The point \(B\) on the curve has polar coordinates \(\quad a ( 2 + \sqrt { 3 } ) , \frac { \pi } { 6 }\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\) and the line \(A B\).
2. Use calculus to determine the exact area of the shaded region \(R\). Give your answer in the form $$\frac { a ^ { 2 } } { 4 } ( d \pi - e + f \sqrt { 3 } )$$ where \(d , e\) and \(f\) are integers.

1. The curve \(C\), with pole \(O\), has polar equation

$$r = 1 + \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the tangent to \(C\) is parallel to the initial line.

1. Find the polar coordinates of \(A\).
2. Find the finite area enclosed by the initial line, the line \(O A\) and the curve \(C\), giving your answer in the form \(a \pi + b \sqrt { 3 }\), where \(a\) and \(b\) are rational constants to be found.

6. \begin{figure}[h] \end{figure} The curve shown in Figure 1 has polar equation $$r = 4 a ( 1 + \cos \theta ) \quad 0 \leqslant \theta < \pi$$ where \(a\) is a positive constant.
The tangent to the curve at the point \(A\) is parallel to the initial line.

1. Show that the polar coordinates of \(A\) are \(\left( 6 a , \frac { \pi } { 3 } \right)\) The point \(B\) lies on the curve such that angle \(A O B = \frac { \pi } { 6 }\) The finite region \(R\), shown shaded in Figure 1, is bounded by the line \(A B\) and the curve.
2. Use calculus to determine the area of the shaded region \(R\), giving your answer in the form \(a ^ { 2 } ( n \pi + p \sqrt { 3 } + q )\), where \(n , p\) and \(q\) are integers.

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$r = 6 ( 1 + \cos \theta ) \quad 0 \leqslant \theta \leqslant \pi$$ Given that \(C\) meets the initial line at the point \(A\), as shown in Figure 1,

1. write down the polar coordinates of \(A\). The line \(l _ { 1 }\), also shown in Figure 1, is the tangent to \(C\) at the point \(B\) and is parallel to the initial line.
2. Use calculus to determine the polar coordinates of \(B\). The line \(l _ { 2 }\), also shown in Figure 1, is the tangent to \(C\) at \(A\) and is perpendicular to the initial line. The region \(R\), shown shaded in Figure 1, is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\)
3. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(p \sqrt { 3 } + q \pi\) where \(p\) and \(q\) are constants 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.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3 \cos \theta , \quad a > 0 , \quad 0 \leqslant \theta < 2 \pi$$ The area enclosed by the curve is \(\frac { 107 } { 2 } \pi\).
Find the value of \(a\).

4. A curve \(C\) has polar equation \(r ^ { 2 } = a ^ { 2 } \cos 2 \theta , 0 \leq \theta \leq \frac { \pi } { 4 }\). The line \(l\) is parallel to the initial line, and \(l\) is the tangent to \(C\) at
above. above.

1. 1. Show that, for any point on \(C , r ^ { 2 } \sin ^ { 2 } \theta\) can be expressed in terms of \(\sin \theta\) and \(a\) only. (1)
   2. Hence, using differentiation, show that the polar coordinates of \(P\) are \(\left( \frac { a } { \sqrt { 2 } } , \frac { \pi } { 6 } \right)\).(6) \includegraphics[max width=\textwidth, alt={}, center]{2352f367-ddf9-4770-ace5-b561b0fbabbb-1_298_725_2163_1169} The shaded region \(R\), shown in the figure above, is bounded by \(C\), the line \(l\) and the half-line with equation \(\theta = \frac { \pi } { 2 }\).
2. Show that the area of \(R\) is \(\frac { a ^ { 2 } } { 16 } ( 3 \sqrt { 3 } - 4 )\).

4. The curve \(C\) has polar equation \(r = 3 a \cos \theta , - \frac { \pi } { 2 } \leq \frac { \pi } { 2 }\). The curve \(D\) has polar equation \(r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\). Given that \(a\) is a positive constant, (a) sketch, on the same diagram, the graphs of \(C\) and \(D\), indicating where each curve cuts the initial line. The graphs of \(C\) intersect at the pole \(O\) and at the points \(P\) and \(Q\).
(b) Find the polar coordinates of \(P\) and \(Q\).
(c) Use integration to find the exact area enclosed by the curve \(D\) and the lines \(\theta = 0\) and \(\theta = \frac { \pi } { 3 }\) 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 $$\frac { 3 a ^ { 2 } } { 16 } ( 2 \pi - 3 \sqrt { } 3 )$$ (d) show that the area of \(R\) is \(\pi a ^ { 2 }\).

8. \section*{Figure 1} The curve \(C\) shown in Fig. 1 has polar equation $$r = a ( 3 + \sqrt { 5 } \cos \theta ) , \quad - \pi \leq \theta < \pi .$$ \includegraphics[max width=\textwidth, alt={}, center]{6d92bf8a-df0d-421c-8246-8160f5921ee6-2_460_792_1503_970}

1. Find the polar coordinates of the points \(P\) and \(Q\) where the tangents to \(C\) are parallel to the initial line. (6) The curve \(C\) represents the perimeter of the surface of a swimming pool. The direct distance from \(P\) to \(Q\) is 20 m.
2. Calculate the value of \(a\).
3. Find the area of the surface of the pool. (6)

7. \begin{figure}[h] \end{figure} A logo is designed which consists of two overlapping closed curves. The polar equations of these curves are \(r = \boldsymbol { a } ( \mathbf { 3 } + \mathbf { 2 } \cos \boldsymbol { \theta } )\) 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 corrdinates of the points \(A\) and \(B\) where the curves meet the initial line.(2)
2. Find the polar coordinates of the points \(\boldsymbol { C }\) and \(\boldsymbol { D }\) where the two curves meet. (4)
3. Show that the area of the overlapping region, which is shaded in the figure, is $$\frac { a ^ { 2 } } { 3 } ( 49 \pi - 48 \sqrt { } 3 )$$

5. (a) Sketch the curve with polar equation \(\quad r = 3 \cos 2 \theta , \quad - \frac { \pi } { 4 } \leq \theta < \frac { \pi } { 4 }\) (b) Find the area of the smaller finite region enclosed between the curve and the half-line $$\theta = \frac { \pi } { 6 }$$ (c) Find the exact distance between the two tangents which are parallel to the initial line.
(8)(Total 16 marks)

9. The diagram is a sketch of the two curves \(C _ { 1 }\) and \(C _ { 2 }\) with polar equations \(C _ { 1 } : r = 3 a ( 1 - \cos \theta ) , - \pi \leq \theta < \pi\) \(\mathrm { C } _ { 2 } : r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\). \includegraphics[max width=\textwidth, alt={}, center]{8646b60a-3822-4d41-8978-1ccad1e216d6-2_318_776_1567_1082} 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 \(B\).
2. Show that the length of the line \(A B\) is \(\frac { 3 \sqrt { } 3 } { 2 } a\). The region inside \(C _ { 2 }\) and outside \(C _ { 1 }\) is shown shaded in the diagram above.
3. Find, in terms of \(a\), the area of this region. A badge is designed which has the shape of the shaded region.
   Given that the length of the line \(A B\) is 4.5 cm ,
4. calculate the area of this badge, giving your answer to three significant figures.
   (Total 16 marks)

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)

8. (a) Sketch the curve \(C\) with polar equation $$r = 5 + \sqrt { 3 } \cos \theta , \quad 0 \leq \theta \leq 2 \pi$$ (b) Find the polar coordinates of the points where the tangents to \(C\) are parallel to the initial line \(\theta = 0\). Give your answers to 3 significant figures where appropriate.
(c) Using integration, find the area enclosed by the curve \(C\), giving your answer in terms of \(\pi\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3 \cos \theta , \quad a > 0 , \quad 0 \leqslant \theta < 2 \pi$$ The area enclosed by the curve is \(\frac { 107 } { 2 } \pi\).
Find the value of \(a\).

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 $$r = 1.5 + \sin 3 \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$

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.

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 \(R\).

8. \begin{figure}[h] \end{figure} Figure 1 shows a closed curve \(C\) with equation $$r = 3 ( \cos 2 \theta ) ^ { \frac { 1 } { 2 } } , \quad \text { where } - \frac { \pi } { 4 } < \theta \leqslant \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } < \theta \leqslant \frac { 5 \pi } { 4 }$$ The lines \(P Q , S R , P S\) and \(Q R\) are tangents to \(C\), where \(P Q\) and \(S R\) are parallel to the initial line and \(P S\) and \(Q R\) are perpendicular to the initial line. The point \(O\) is the pole.

1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1.
2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1.

9. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curves given by the polar equations $$r = 1 \text { and } r = 2 - 2 \sin \theta$$

1. Find the coordinates of the points where the curves intersect. The region \(S\), between the curves, for which \(r < 1\) and for which \(r < 2 - 2 \sin \theta\), 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 rational numbers.

8. \begin{figure}[h] \end{figure} Figure 1 shows a curve \(C\) with polar equation \(r = a \sin 2 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\), and a half-line \(l\).
The half-line \(l\) meets \(C\) at the pole \(O\) and at the point \(P\). The tangent to \(C\) at \(P\) is parallel to the initial line. The polar coordinates of \(P\) are \(( R , \phi )\).

1. Show that \(\cos \phi = \frac { 1 } { \sqrt { 3 } }\)
2. Find the exact value of \(R\). The region \(S\), shown shaded in Figure 1, is bounded by \(C\) and \(l\).
3. Use calculus to show that the exact area of \(S\) is $$\frac { 1 } { 36 } a ^ { 2 } \left( 9 \arccos \left( \frac { 1 } { \sqrt { 3 } } \right) + \sqrt { 2 } \right)$$