Area enclosed by polar curve

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 \(\theta = \frac { \pi } { 2 }\).

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 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. 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, \(\int _ { - h } ^ { h } x \arctan ( \sqrt { 3 } + x ) \mathrm { d } x \approx \frac { 1 } { 6 } h ^ { 3 }\).

1

1. A curve has polar equation \(r = 1 + \cos \theta\) for \(0 \leqslant \theta < 2 \pi\).
   1. Sketch the curve.
   2. Find the area of the region enclosed by the curve, giving your answer in exact form.
2. Assuming that \(x ^ { 4 }\) and higher powers may be neglected, write down the Maclaurin series approximations for \(\sin x\) and \(\cos x\) (where \(x\) is in radians). Hence or otherwise obtain an approximation for \(\tan x\) in the form \(a x + b x ^ { 3 }\).
3. Find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 - \frac { 1 } { 4 } X ^ { 2 } } } \mathrm {~d} x\), giving your answer in exact form.

1

1. A curve has polar equation \(r = a ( 1 - \sin \theta )\), where \(a > 0\) and \(0 \leqslant \theta < 2 \pi\).
   1. Sketch the curve.
   2. Find, in an exact form, the area of the region enclosed by the curve.
2. 1. Find, in an exact form, the value of the integral \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { 1 + 4 x ^ { 2 } } \mathrm {~d} x\).
   2. Find, in an exact form, the value of the integral \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 + 4 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\).

7 A curve has polar equation \(r = 1 + \cos 3 \theta\), for \(- \pi < \theta \leqslant \pi\).

1. Show that the line \(\theta = 0\) is a line of symmetry.
2. Find the equations of the tangents at the pole.
3. Find the exact value of the area of the region enclosed by the curve between \(\theta = - \frac { 1 } { 3 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\).

8 A curve has polar equation \(r = a ( 1 + \cos \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\).

1. Find the equation of the tangent at the pole.
2. Sketch the curve.
3. Find the area enclosed by the curve.

9 The equation of a curve in polar coordinates is \(r = 2 \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).

1. Sketch the curve.
2. Find the area of the region enclosed by this curve.
3. By expressing \(\sin 3 \theta\) in terms of \(\sin \theta\), show that a cartesian equation for the curve is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } y - 2 y ^ { 3 } .$$ \section*{END OF QUESTION PAPER}

5 The curve \(C\) has polar equation \(r = 2 \cos 2 \theta\). Sketch the curve for \(0 \leqslant \theta < 2 \pi\). Find the exact area of one loop of the curve.

1 Find the area of the region enclosed by the curve with polar equation \(r = 2 ( 1 + \cos \theta )\), for \(0 \leqslant \theta < 2 \pi\).

10 Use the identity \(2 \sin P \cos Q \equiv \sin ( P + Q ) + \sin ( P - Q )\) to show that $$2 \sin \theta \cos \left( \theta - \frac { 1 } { 4 } \pi \right) \equiv \cos \left( 2 \theta - \frac { 3 } { 4 } \pi \right) + \frac { 1 } { \sqrt { } 2 }$$ A curve has polar equation \(r = 2 \sin \theta \cos \left( \theta - \frac { 1 } { 4 } \pi \right)\), for \(0 \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\). Sketch the curve and state the polar equation of its line of symmetry, justifying your answer. Show that the area of the region enclosed by the curve is \(\frac { 3 } { 8 } ( \pi + 1 )\).

4 The curve \(C\) has cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 a ^ { 2 } x y\), where \(a\) is a positive constant. Show that the polar equation of \(C\) is \(r ^ { 2 } = a ^ { 2 } \sin 2 \theta\). Sketch \(C\) for \(- \pi < \theta \leqslant \pi\). Find the area enclosed by one loop of \(C\).

11 The curve \(C\) has polar equation \(r = a ( 1 + \sin \theta )\) for \(- \pi < \theta \leqslant \pi\), where \(a\) is a positive constant.

1. Sketch \(C\).
2. Find the area of the region enclosed by \(C\).
3. Show that the length of the arc of \(C\) from the pole to the point furthest from the pole is given by $$s = ( \sqrt { } 2 ) a \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \sqrt { } ( 1 + \sin \theta ) \mathrm { d } \theta$$
4. Show that the substitution \(u = 1 + \sin \theta\) reduces this integral for \(s\) to \(( \sqrt { } 2 ) a \int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { } ( 2 - u ) } \mathrm { d } u\). Hence evaluate \(s\).

A curve \(C\) has polar equation \(r = 2 a \cos \left( 2 \theta + \frac { 1 } { 2 } \pi \right)\) for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.

1. Show that \(r = - 2 a \sin 2 \theta\) and sketch \(C\).
2. Deduce that the cartesian equation of \(C\) is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = - 4 a x y .$$
3. Find the area of one loop of \(C\).
4. Show that, at the points (other than the pole) at which a tangent to \(C\) is parallel to the initial line, $$2 \tan \theta = - \tan 2 \theta .$$

3 The curve \(C\) has polar equation \(r = \cos 2 \theta\), for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).

1. Sketch \(C\).
2. Find the area of the region enclosed by \(C\), showing full working.
3. Find a cartesian equation of \(C\).

The polar equation of a curve \(C\) is \(r = a ( 1 + \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.

1. Sketch \(C\).
2. Show that the cartesian equation of \(C\) is $$x ^ { 2 } + y ^ { 2 } = a \left( x + \sqrt { } \left( x ^ { 2 } + y ^ { 2 } \right) \right)$$
3. Find the area of the sector of \(C\) between \(\theta = 0\) and \(\theta = \frac { 1 } { 3 } \pi\).
4. Find the arc length of \(C\) between the point where \(\theta = 0\) and the point where \(\theta = \frac { 1 } { 3 } \pi\).

3 The curve \(C\) has polar equation \(r = a \cos 3 \theta\), for \(- \frac { 1 } { 6 } \pi \leqslant \theta \leqslant \frac { 1 } { 6 } \pi\), where \(a\) is a positive constant.

1. Sketch \(C\).
2. Find the area of the region enclosed by \(C\), showing full working.
3. Using the identity \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\), find a cartesian equation of \(C\).

7 The curve \(C\) has polar equation $$r = \theta \sin \theta ,$$ where \(0 \leqslant \theta \leqslant \pi\). Draw a sketch of \(C\). Find the area of the region enclosed by \(C\), leaving your answer in terms of \(\pi\).

3 The diagram shows a sketch of a loop, the pole \(O\) and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{f4fdffc7-5647-4462-a983-1564d4e76a4d-3_305_553_383_740} The polar equation of the loop is $$r = ( 2 + \cos \theta ) \sqrt { \sin \theta } , \quad 0 \leqslant \theta \leqslant \pi$$ Find the area enclosed by the loop.

6 The diagram shows a sketch of a curve \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{8cb4b110-274e-47ec-a31b-ee8f84434a65-3_305_556_1078_721} The polar equation of the curve is $$r = 2 \sin 2 \theta \sqrt { \cos \theta } , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ Show that the area of the region bounded by \(C\) is \(\frac { 16 } { 15 }\).

8 The diagram shows a sketch of the curve \(C\) with polar equation $$r = 3 + 2 \cos \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{80c4336c-b0ca-46de-a871-812e6923f7f2-4_463_668_1468_699}

1. Find the area of the region bounded by the curve \(C\).
2. A circle, whose cartesian equation is \(( x - 4 ) ^ { 2 } + y ^ { 2 } = 16\), intersects the curve \(C\) at the points \(A\) and \(B\).
   1. Find, in surd form, the length of \(A B\).
   2. Find the perimeter of the segment \(A O B\) of the circle, where \(O\) is the pole.

4 The diagram shows the curve \(C\) with polar equation $$r = 6 ( 1 - \cos \theta ) , \quad 0 \leqslant \theta < 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{06ae13de-5cf3-421d-ac7a-ee9f74b653be-3_552_903_922_550}

1. Find the area of the region bounded by the curve \(C\).
2. The circle with cartesian equation \(x ^ { 2 } + y ^ { 2 } = 9\) intersects the curve \(C\) at the points \(A\) and \(B\).
   1. Find the polar coordinates of \(A\) and \(B\).
   2. Find, in surd form, the length of \(A B\).