4.09c Area enclosed: by polar curve

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 \(C\).

8 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations, for \(0 \leqslant \theta \leqslant \pi\), as follows: $$\begin{aligned} & C _ { 1 } : r = a \\ & C _ { 2 } : r = 2 a | \cos \theta | \end{aligned}$$ where \(a\) is a positive constant. The curves intersect at the points \(P _ { 1 }\) and \(P _ { 2 }\).

1. Find the polar coordinates of \(P _ { 1 }\) and \(P _ { 2 }\).
2. In a single diagram, sketch \(C _ { 1 } , C _ { 2 }\) and their line of symmetry.
3. The region \(R\) enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) is bounded by the \(\operatorname { arcs } O P _ { 1 } , P _ { 1 } P _ { 2 }\) and \(P _ { 2 } O\), where \(O\) is the pole. Find the area of \(R\), giving your answer in exact form.

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

1. The point on \(C _ { 1 }\) furthest from the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(P\). Show that, at \(P\), $$2 \theta \tan \theta = 1$$ and verify that this equation has a root between 0.6 and 0.7 .
   The curve \(C _ { 2 }\) has polar equation \(r ^ { 2 } = \theta \sec ^ { 2 } \theta\), for \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole, denoted by \(O\), and at another point \(Q\).
2. Find the exact value of \(\theta\) at \(Q\).
3. The diagram below shows the curve \(C _ { 2 }\). Sketch \(C _ { 1 }\) on this diagram.
4. Find, in exact form, the area of the region \(O P Q\) enclosed by \(C _ { 1 }\) and \(C _ { 2 }\).

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 $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \sqrt { } \left( 1 + \theta ^ { 2 } \right) } { \theta ^ { 2 } } \mathrm {~d} \theta$$

4 The curve \(C\) has polar equation $$r = \mathrm { e } ^ { \frac { 1 } { 5 } \theta } , \quad 0 \leqslant \theta \leqslant \frac { 3 } { 2 } \pi$$

1. Draw a sketch of \(C\).
2. Find the length of \(C\), correct to 3 significant figures.

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 \(\pi\).

10 The curve \(C\) has polar equation $$r = a \sin 3 \theta$$ where \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).

1. Show that the area of the region enclosed by \(C\) is \(\frac { 1 } { 12 } \pi a ^ { 2 }\).
2. Show that, at the point of \(C\) at maximum distance from the initial line, $$\tan 3 \theta + 3 \tan \theta = 0 .$$
3. Use the formula $$\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$ to find this maximum distance.
4. Draw a sketch of \(C\).

8 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations given by $$\begin{array} { l l r } C _ { 1 } : & r = 3 \sin \theta , & 0 \leqslant \theta < \pi , \\ C _ { 2 } : & r = 1 + \sin \theta , & - \pi < \theta \leqslant \pi . \end{array}$$

1. Find the polar coordinates of the points, other than the pole, where \(C _ { 1 }\) and \(C _ { 2 }\) meet.
2. In a single diagram, draw sketch graphs of \(C _ { 1 }\) and \(C _ { 2 }\).
3. Show that the area of the region which is inside \(C _ { 1 }\) but outside \(C _ { 2 }\) is \(\pi\).

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.

10 The curve \(C\) has polar equation \(r = 2 \sin \theta ( 1 - \cos \theta )\), for \(0 \leqslant \theta \leqslant \pi\). Find \(\frac { \mathrm { d } r } { \mathrm {~d} \theta }\) and hence find the polar coordinates of the point of \(C\) that is furthest from the pole. Sketch \(C\). Find the exact area of the sector from \(\theta = 0\) to \(\theta = \frac { 1 } { 4 } \pi\).

8 A circle has polar equation \(r = a\), for \(0 \leqslant \theta < 2 \pi\), and a cardioid has polar equation \(r = a ( 1 - \cos \theta )\), for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant. Draw sketches of the circle and the cardioid on the same diagram. Write down the polar coordinates of the points of intersection of the circle and the cardioid. Show that the area of the region that is both inside the circle and inside the cardioid is $$\left( \frac { 5 } { 4 } \pi - 2 \right) a ^ { 2 }$$

A curve \(C\) has parametric equations $$x = 1 - 3 t ^ { 2 } , \quad y = t \left( 1 - 3 t ^ { 2 } \right) , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }$$ Show that \(\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + 9 t ^ { 2 } \right) ^ { 2 }\). Hence find

1. the arc length of \(C\),
2. the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Use the fact that \(t = \frac { y } { x }\) to find a cartesian equation of \(C\). Hence show that the polar equation of \(C\) is \(r = \sec \theta \left( 1 - 3 \tan ^ { 2 } \theta \right)\), and state the domain of \(\theta\). Find the area of the region enclosed between \(C\) and the initial line. {www.cie.org.uk} after the live examination series. }

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 \(\theta = \frac { 1 } { 3 } \pi\).

The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations, for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), as follows: $$\begin{aligned} & C _ { 1 } : r = 2 \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right) , \\ & C _ { 2 } : r = \mathrm { e } ^ { 2 \theta } - \mathrm { e } ^ { - 2 \theta } \end{aligned}$$ The curves intersect at the point \(P\) where \(\theta = \alpha\).

1. Show that \(\mathrm { e } ^ { 2 \alpha } - 2 \mathrm { e } ^ { \alpha } - 1 = 0\). Hence find the exact value of \(\alpha\) and show that the value of \(r\) at \(P\) is \(4 \sqrt { } 2\).
2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
3. Find the area of the region enclosed by \(C _ { 1 } , C _ { 2 }\) and the initial line, giving your answer correct to 3 significant figures.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\).

2 The curve \(C\) has polar equation \(r = \mathrm { e } ^ { 4 \theta }\) for \(0 \leqslant \theta \leqslant \alpha\), where \(\alpha\) is measured in radians. The length of \(C\) is 2015 . Find the value of \(\alpha\).

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 \(\pi\).

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 \(C\).