1.08f Area between two curves: using integration

4 In this question you must show detailed reasoning. \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_646_812_312_242} The diagram shows parts of the curves \(y = 3 \sqrt { x }\) and \(y = 4 - x ^ { 2 }\), which intersect at the point ( 1,3 ). The shaded region, bounded by the two curves and the \(y\)-axis, is occupied by a uniform lamina. Determine the exact \(x\)-coordinate of the centre of mass of the lamina.

1. (a) Given

$$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad n \in \mathbb { N }$$ show that $$32 \cos ^ { 6 } \theta \equiv \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10$$ \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows a solid paperweight with a flat base.
Figure 2 shows the curve with equation $$y = H \cos ^ { 3 } \left( \frac { x } { 4 } \right) \quad - 4 \leqslant x \leqslant 4$$ where \(H\) is a positive constant and \(x\) is in radians.
The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = - 4\), the line with equation \(x = 4\) and the \(x\)-axis. The paperweight is modelled by the solid of revolution formed when \(R\) is rotated \(\mathbf { 1 8 0 } ^ { \circ }\) about the \(x\)-axis. Given that the maximum height of the paperweight is 2 cm ,
(b) write down the value of \(H\).
(c) Using algebraic integration and the result in part (a), determine, in \(\mathrm { cm } ^ { 3 }\), the volume of the paperweight, according to the model. Give your answer to 2 decimal places.
[0pt] [Solutions based entirely on calculator technology are not acceptable.]
(d) State a limitation of the model.

14 In this question you must show detailed reasoning. Fig. 14 shows the graphs of \(y = \sin x \cos 2 x\) and \(y = \frac { 1 } { 2 } - \sin 2 x \cos x\). \begin{figure}[h] \end{figure} Use integration to find the area between the two curves, giving your answer in an exact form.

6 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-06_463_702_264_685} The diagram shows the curve \(C\) with parametric equations $$x = \frac { 1 } { 4 } \sin t , \quad y = t \cos t$$ where \(0 \leqslant t \leqslant k\).

1. Find the value of \(k\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) in terms of \(t\). The maximum point on \(C\) is denoted by \(P\).
3. Using your answer to part (ii) and the standard small angle approximations, find an approximation for the \(x\)-coordinate of \(P\).
4. (a) Show that the area of the finite region bounded by \(C\) and the \(x\)-axis is given by $$b \int _ { 0 } ^ { a } t ( 1 + \cos 2 t ) \mathrm { d } t$$ where \(a\) and \(b\) are constants to be determined.
   (b) In this question you must show detailed reasoning. Hence find the exact area of the finite region bounded by \(C\) and the \(x\)-axis.

6 The curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-5_428_563_372_740} The curve cuts the \(x\)-axis at the point \(A ( - 1,0 )\) and cuts the \(y\)-axis at the point \(B\).

1. 1. State the coordinates of the point \(B\) and hence find the area of the triangle \(A O B\), where \(O\) is the origin.
   2. Find \(\int \left( 3 x ^ { 5 } + 2 x + 5 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve and the line \(A B\).
2. 1. Find the gradient of the curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) at the point \(A ( - 1,0 )\).
   2. Hence find an equation of the tangent to the curve at the point \(A\).

6

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\).
   1. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\) as the product of three linear factors.
2. The curve with equation \(y = x ^ { 3 } - 7 x - 6\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{de4f827d-f237-488a-9177-3d85d0cb1771-4_403_762_651_641} The curve cuts the \(x\)-axis at the point \(A\) and the points \(B ( - 1,0 )\) and \(C ( 3,0 )\).
   1. State the coordinates of the point \(A\).
   2. Find \(\int _ { - 1 } ^ { 3 } \left( x ^ { 3 } - 7 x - 6 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - 7 x - 6\) and the \(x\)-axis between \(B\) and \(C\).
   4. Find the gradient of the curve \(y = x ^ { 3 } - 7 x - 6\) at the point \(B\).
   5. Hence find an equation of the normal to the curve at the point \(B\).

6

1. The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 4 x - 5\).
   1. Use the Factor Theorem to show that \(x - 1\) is a factor of \(\mathrm { f } ( x )\).
   2. Express \(\mathrm { f } ( x )\) in the form \(( x - 1 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
   3. Hence show that the equation \(\mathrm { f } ( x ) = 0\) has exactly one real root and state its value.
2. The curve with equation \(y = x ^ { 3 } + 4 x - 5\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{23f34515-3373-4644-a8a1-82b45809d934-4_505_959_868_529} The curve cuts the \(x\)-axis at the point \(A ( 1,0 )\) and the point \(B ( 2,11 )\) lies on the curve.
   1. Find \(\int \left( x ^ { 3 } + 4 x - 5 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(A B\).

5 The curve with equation \(y = 16 - x ^ { 4 }\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-3_435_663_824_685} The points \(A ( - 2,0 ) , B ( 2,0 )\) and \(C ( 1,15 )\) lie on the curve.

1. Find an equation of the straight line \(A C\).
2. 1. Find \(\int _ { - 2 } ^ { 1 } \left( 16 - x ^ { 4 } \right) \mathrm { d } x\).
   2. Hence calculate the area of the shaded region bounded by the curve and the line \(A C\).

4

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\).
   1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
   2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   3. Express \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   4. The equation \(\mathrm { p } ( x ) = 0\) has one root equal to - 2 . Show that the equation has no other real roots.
2. The curve with equation \(y = x ^ { 3 } - x + 6\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{5f1ff5fa-b6e8-4c4f-aef7-63eb947b299f-3_529_702_945_667} The curve cuts the \(x\)-axis at the point \(A ( - 2,0 )\) and the \(y\)-axis at the point \(B\).
   1. State the \(y\)-coordinate of the point \(B\).
   2. Find \(\int _ { - 2 } ^ { 0 } \left( x ^ { 3 } - x + 6 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - x + 6\) and the line \(A B\).

8 In this question you must show detailed reasoning. A is the point \(( 1,0 ) , B\) is the point \(( 1,1 )\) and \(D\) is the point where the tangent to the curve \(y = x ^ { 3 }\) at B crosses the \(x\)-axis, as shown in Fig. 8. The tangent meets the \(y\)-axis at E. \begin{figure}[h] \end{figure}

1. Find the area of triangle ODE.
2. Find the area of the region bounded by the curve \(y = x ^ { 3 }\), the tangent at B and the \(y\)-axis.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = x ^ { 2 } + 3 \quad x > 0$$ and part of the curve \(C _ { 2 }\) with equation $$y = 13 - \frac { 9 } { x ^ { 2 } } \quad x > 0$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(P\) and \(Q\) as shown in Figure 1 .

1. Use algebra to find the \(x\) coordinate of \(P\) and the \(x\) coordinate of \(Q\). The finite region \(R\), shown shaded in Figure 1, is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
2. Use algebraic integration to find the exact area of \(R\).

9

1. On the same axes, sketch the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).
2. Find the exact area of the region contained between the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).

10 \includegraphics[max width=\textwidth, alt={}, center]{a3cad2ad-e06b-4aa4-a3a9-a2840cd54893-3_529_527_264_810} The diagram shows the region \(R\) in the first quadrant bounded by the curves \(y = \frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right)\) and \(y = \frac { 1 } { 5 } \left( 9 - x ^ { 2 } \right)\). \(R\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Calculate the volume of the solid formed.

5 \includegraphics[max width=\textwidth, alt={}, center]{7895dcbc-2ae0-498f-8770-7b738feed7c9-2_746_1182_1304_479} The diagram shows the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 13 x + 10\) and the tangent to the curve at the point ( 2,0 ).

1. Find the equation of this tangent and verify that the tangent intersects the curve when \(x = - 6\).
2. Calculate the exact area of the region bounded by the curve and the tangent.

5 \includegraphics[max width=\textwidth, alt={}, center]{69214874-18a7-495d-892d-2a0a7019cbe9-2_746_1182_1304_479} The diagram shows the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 13 x + 10\) and the tangent to the curve at the point ( 2,0 ).

1. Find the equation of this tangent and verify that the tangent intersects the curve when \(x = - 6\).
2. Calculate the exact area of the region bounded by the curve and the tangent.

\includegraphics{figure_9} The diagram shows the curves with equations \(y = x^3 - 3x + 3\) and \(y = 2x^3 - 4x^2 + 3\).

1. Find the \(x\)-coordinates of the points of intersection of the curves. [3]
2. Find the area of the shaded region. [4]

\includegraphics{figure_9} The diagram shows the curve \(y = (x - 2)^2\) and the line \(y + 2x = 7\), which intersect at points \(A\) and \(B\). Find the area of the shaded region. [8]

\includegraphics{figure_2} The line with equation \(y = 3x + 20\) cuts the curve with equation \(y = x^2 + 6x + 10\) at the points \(A\) and \(B\), as shown in Figure 2.

1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). [5]

The shaded region \(S\) is bounded by the line and the curve, as shown in Figure 2.

1. Use calculus to find the exact area of \(S\). [7]

\includegraphics{figure_1} Figure 1 shows part of a curve \(C\) with equation \(y = 2x + \frac{8}{x^2} - 5\), \(x > 0\). The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 4 respectively. The region \(R\), shaded in Figure 1, is bounded by \(C\) and the straight line joining \(P\) and \(Q\).

1. Find the exact area of \(R\). [8]
2. Use calculus to show that \(y\) is increasing for \(x > 2\). [4]

\includegraphics{figure_2} In Figure 2 the curve \(C\) has equation \(y = 6x - x^2\) and the line \(L\) has equation \(y = 2x\).

1. Show that the curve \(C\) intersects the \(x\)-axis at \(x = 0\) and \(x = 6\). [1]
2. Show that the line \(L\) intersects the curve \(C\) at the points \((0, 0)\) and \((4, 8)\). [3]

The region \(R\), bounded by the curve \(C\) and the line \(L\), is shown shaded in Figure 2.

1. Use calculus to find the area of \(R\). [6]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 5 + 2x - x^2\) and the line with equation \(y = 2\). The curve and the line intersect at the points \(A\) and \(B\).

1. Find the \(x\)-coordinates of \(A\) and \(B\). [3]

The shaded region \(R\) is bounded by the curve and the line.

1. Find the area of \(R\). [6]

\includegraphics{figure_7} Figure 2 shows the line with equation \(y = x + 1\) and the curve with equation \(y = 6x - x^2 - 3\). The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.

1. Calculate the coordinates of \(A\) and the coordinates of \(B\). [5]

The shaded region \(R\) is bounded by the line and the curve.

1. Calculate the area of \(R\). [7]