1.08d Evaluate definite integrals: between limits

4

1. A curve is defined by the parametric equations \(x = 3 \cos 2 \theta , y = 2 \cos \theta\).
   1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { k \cos \theta }\), where \(k\) is an integer.
   2. Find an equation of the normal to the curve at the point where \(\theta = \frac { \pi } { 3 }\).
2. Find the exact value of \(\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \sin ^ { 2 } x \mathrm {~d} x\).

1

1. 1. Express \(\frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) }\) in the form \(\frac { A } { 2 + x } + \frac { B } { 1 - 3 x }\), where \(A\) and \(B\) are integers.
      (3 marks)
   2. Hence show that \(\int _ { - 1 } ^ { 0 } \frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) } \mathrm { d } x = p \ln 2\), where \(p\) is rational.
      (4 marks)
2. 1. Given that \(\frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }\) can be written as \(C + \frac { 5 - 8 x } { 2 - 5 x - 3 x ^ { 2 } }\), find the value of \(C\).
      (1 mark)
   2. Hence find the exact value of the area of the region bounded by the curve \(y = \frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }\), the \(x\)-axis and the lines \(x = - 1\) and \(x = 0\). You may assume that \(y > 0\) when \(- 1 \leqslant x \leqslant 0\).

1 It is given that \(\mathrm { f } ( x ) = \frac { 19 x - 2 } { ( 5 - x ) ( 1 + 6 x ) }\) can be expressed as \(\frac { A } { 5 - x } + \frac { B } { 1 + 6 x }\), where \(A\) and \(B\) are integers.

1. Find the values of \(A\) and \(B\).
2. Hence show that \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = k \ln 5\), where \(k\) is a rational number.
   [0pt] [6 marks]

5. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \frac { 1 } { \sqrt { 3 x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 5\).

1. Find the area of the shaded region. The shaded region is rotated completely about the \(x\)-axis.
2. Find the volume of the solid formed, giving your answer in the form \(k \pi \ln 2\), where \(k\) is a simplified fraction.
   5. continued

3. $$f ( x ) = \frac { 7 + 3 x + 2 x ^ { 2 } } { ( 1 - 2 x ) ( 1 + x ) ^ { 2 } } , \quad | x | > \frac { 1 } { 2 }$$

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that $$\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = p - \ln q$$ where \(p\) is rational and \(q\) is an integer.
   3. continued

2 The speed of a 100 metre runner in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) is measured electronically every 4 seconds.
The measurements are plotted as points on the speed-time graph in Fig. 6. The vertical dotted line is drawn through the runner's finishing time. Fig. 6 also illustrates Model P in which the points are joined by straight lines. \begin{figure}[h] \end{figure}

1. Use Model P to estimate
   (A) the distance the runner has gone at the end of 12 seconds,
   (B) how long the runner took to complete 100 m . A mathematician proposes Model Q in which the runner's speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\), is given by $$v = \frac { 5 } { 2 } t - \frac { 1 } { 8 } t ^ { 2 } .$$
2. Verify that Model Q gives the correct speed for \(t = 8\).
3. Use Model Q to estimate the distance the runner has gone at the end of 12 seconds.
4. The runner was timed at 11.35 seconds for the 100 m . Which model places the runner closer to the finishing line at this time? In this question take \(g\) as \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   A small ball is released from rest. It falls for 2 seconds and is then brought to rest over the next 5 seconds. This motion is modelled in the speed-time graph Fig. 6. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4f80ea36-001f-4a00-849f-542f5072516b-3_658_1101_281_503} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure} For this model,
5. calculate the distance fallen from \(t = 0\) to \(t = 7\),
6. find the acceleration of the ball from \(t = 2\) to \(t = 6\), specifying the direction,
7. obtain an expression in terms of \(t\) for the downward speed of the ball from \(t = 2\) to \(t = 6\),
8. state the assumption that has been made about the resistance to motion from \(t = 0\) to \(t = 2\). The part of the motion from \(t = 2\) to \(t = 7\) is now modelled by \(v = - \frac { 3 } { 2 } t ^ { 2 } + \frac { 19 } { 2 } t + 7\).
9. Verify that \(v\) agrees with the values given in Fig, 6 at \(t = 2 , t = 6\) and \(t = 7\).
10. Calculate the distance fallen from \(t = 2\) to \(t = 7\) according to this model.

5 Fig. 3 is a sketch of the velocity-time graph modelling the velocity of a sprinter at the start of a race. \begin{figure}[h] \end{figure}

1. How can you tell from the sketch that the acceleration is not modelled as being constant for \(0 \leqslant t \leqslant 4\) ? The velocity of the sprinter, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), for the time interval \(0 \leqslant t \leqslant 4\) is modelled by the expression $$v = 3 t - \frac { 3 } { 8 } t ^ { 2 } .$$
2. Find the acceleration that the model predicts for \(t = 4\) and comment on what this suggests about the running of the sprinter.
3. Calculate the distance run by the sprinter from \(t = 1\) to \(t = 4\).

2 Fig. 2 shows an acceleration-time graph modelling the motion of a particle. \begin{figure}[h] \end{figure} At \(t = 0\) the particle has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive direction.

1. Find the velocity of the particle when \(t = 2\).
2. At what time is the particle travelling in the negative direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ?

4 A particle has mass 200 kg and moves on a smooth horizontal plane. A single horizontal force, \(\left( 400 \cos \left( \frac { \pi } { 2 } t \right) \mathbf { i } + 600 t ^ { 2 } \mathbf { j } \right)\) newtons, acts on the particle at time \(t\) seconds. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find the acceleration of the particle at time \(t\).
2. When \(t = 4\), the velocity of the particle is \(( - 3 \mathbf { i } + 56 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
3. Find \(t\) when the particle is moving due west.
4. Find the speed of the particle when it is moving due west.
   
   \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-09_2484_1709_223_153}

3 A particle moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), is given by $$\mathbf { v } = 4 \mathrm { e } ^ { - 2 t } \mathbf { i } + \left( 6 t - 3 t ^ { 2 } \right) \mathbf { j }$$

1. Find an expression for the acceleration of the particle at time \(t\).
2. The mass of the particle is 5 kg .
   1. Find an expression for the force \(\mathbf { F }\) acting on the particle at time \(t\).
   2. Find the magnitude of \(\mathbf { F }\) when \(t = 0\).
3. Find the value of \(t\) when \(\mathbf { F }\) acts due west.
4. When \(t = 0\), the particle is at the point with position vector \(( 6 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).

2 A particle moves in a straight line. At time \(t\) seconds, it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 6 t ^ { 2 } - 2 \mathrm { e } ^ { - 4 t } + 8$$ and \(t \geqslant 0\).

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. Find the acceleration of the particle when \(t = 0.5\).
2. The particle has mass 4 kg . Find the magnitude of the force acting on the particle when \(t = 0.5\).
3. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).

1 An ice-hockey player has mass 60 kg . He slides in a straight line at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on the horizontal smooth surface of an ice rink towards the vertical perimeter wall of the rink, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-02_476_594_769_715} The player collides directly with the wall, and remains in contact with the wall for 0.5 seconds. At time \(t\) seconds after coming into contact with the wall, the force exerted by the wall on the player is \(4 \times 10 ^ { 4 } t ^ { 2 } ( 1 - 2 t )\) newtons, where \(0 \leqslant t \leqslant 0.5\).

1. Find the magnitude of the impulse exerted by the wall on the player.
2. The player rebounds from the wall. Find the player's speed immediately after the collision.

1 A stone, of mass 2 kg , is moving in a straight line on a smooth horizontal sheet of ice under the action of a single force which acts in the direction of motion. At time \(t\) seconds, the force has magnitude \(( 3 t + 1 )\) newtons, \(0 \leqslant t \leqslant 3\). When \(t = 0\), the stone has velocity \(1 \mathrm {~ms} ^ { - 1 }\).
When \(t = T\), the stone has velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(T\).
(6 marks)

3 A particle of mass 0.5 kg is moving in a straight line on a smooth horizontal surface.
The particle is then acted on by a horizontal force for 3 seconds. This force acts in the direction of motion of the particle and at time \(t\) seconds has magnitude \(( 3 t + 1 ) \mathrm { N }\). When \(t = 0\), the velocity of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the magnitude of the impulse of the force on the particle between the times \(t = 0\) and \(t = 3\).
2. Hence find the velocity of the particle when \(t = 3\).
3. Find the value of \(t\) when the velocity of the particle is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3. At time \(t\) seconds the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of a particle is given by $$a = \frac { 4 } { ( 1 + t ) ^ { 3 } }$$ When \(t = 0\), the particle has velocity \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and displacement 3 m from a fixed origin \(O\).

1. Find an expression for the velocity of the particle in terms of \(t\).
2. Show that when \(t = 3\) the particle is 10.5 m from \(O\).

2

1. Find, in terms of \(p\) and \(q\), the value of the integral \(\int _ { p } ^ { q } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\).
2. Show that only one of the following improper integrals has a finite value, and find that value:
   1. \(\int _ { 0 } ^ { 2 } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\);
   2. \(\int _ { 2 } ^ { \infty } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\).

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where

$$v = 5 \sin 2 t$$ When \(t = 0 , x = 1\) and \(P\) is at rest.

1. Find the magnitude and direction of the acceleration of \(P\) at the instant when \(P\) is next at rest.
2. Show that \(1 \leqslant x \leqslant 6\)
3. Find the total time, in the first \(4 \pi\) seconds of the motion, for which \(P\) is more than 3 metres from \(O\)
   \includegraphics[max width=\textwidth, alt={}]{a7901165-1679-4d30-9444-0c27020e32ea-16_2260_52_309_1982}

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing, where

$$v = \frac { 1 } { 2 } \left( 3 \mathrm { e } ^ { 2 t } - 1 \right) \quad t \geqslant 0$$ The acceleration of \(P\) at time \(t\) seconds is \(a \mathrm {~ms} ^ { - 2 }\)

1. Show that \(a = 2 v + 1\)
2. Find the acceleration of \(P\) when \(t = 0\)
3. Find the exact distance travelled by \(P\) in accelerating from a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. A particle \(P\) is moving along the \(x\)-axis.

At time \(t\) seconds, \(t \geqslant 0 , P\) has acceleration \(a \mathrm {~ms} ^ { - 2 }\) and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where $$v = \mathrm { e } ^ { 2 t } + 6 \mathrm { e } ^ { t } - k t$$ and \(k\) is a positive constant.
When \(t = \ln 2\), \(a = 0\)

1. Find the value of \(k\). When \(t = 0\), the particle passes through the fixed point \(A\).
   When \(t = \ln 2\), the particle is \(d\) metres from \(A\).
2. Showing all stages of your working, find the value of \(d\) correct to 2 significant figures.
   [0pt] [Solutions relying entirely on calculator technology are not acceptable.]

1. A particle \(P\) is moving along the \(x\)-axis. At time \(t\) seconds, \(P\) has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction and acceleration \(a \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction.

In a model of the motion of \(P\) $$a = 4 - 3 v$$ When \(t = 0 , v = 0\)

1. Use integration to show that \(v = k \left( 1 - \mathrm { e } ^ { - 3 t } \right)\), where \(k\) is a constant to be found. When \(t = 0 , P\) is at the origin \(O\)
2. Find, in terms of \(t\) only, the distance of \(P\) from \(O\) at time \(t\) seconds.

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where

$$v = ( t - 2 ) ( 3 t - 10 ) , \quad t \geqslant 0$$ When \(t = 0 , P\) is at the origin \(O\).

1. Find the acceleration of \(P\) at time \(t\) seconds.
2. Find the total distance travelled by \(P\) in the first 2 seconds of its motion.
3. Show that \(P\) never returns to \(O\), explaining your reasoning.
   

10. \begin{figure}[h] \end{figure} A solid playing piece for a board game is modelled by rotating the curve \(C\), shown in Figure 2, through \(2 \pi\) radians about the \(x\)-axis. The curve \(C\) has equation $$y = \sqrt { 1 + \frac { x ^ { 2 } } { 9 } } \quad - 4 \leqslant x \leqslant 4$$ with units as centimetres.

1. Show that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the playing piece is given by $$S = p \pi \int _ { - 4 } ^ { 4 } \sqrt { 81 + 10 x ^ { 2 } } \mathrm {~d} x + q \pi$$ where \(p\) and \(q\) are constants to be determined. Using the substitution \(x = \frac { 9 } { \sqrt { 10 } } \sinh u\), or another algebraic integration method, and showing all your working,
2. determine the total surface area of the playing piece, giving your answer to the nearest \(\mathrm { cm } ^ { 2 }\)

7. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$

1. Prove that, for \(n \geqslant 2\), $$n I _ { n } = ( n - 1 ) I _ { n - 2 }$$
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1c262813-4160-4eda-9a36-e4ba38182c8a-22_588_1018_630_520} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} A designer is asked to produce a poster to completely cover the curved surface area of a solid cylinder which has diameter 1 m and height 0.7 m . He uses a large sheet of paper with height 0.7 m and width of \(\pi \mathrm { m }\).
   Figure 2 shows the first stage of the design, where the poster is divided into two sections by a curve. The curve is given by the equation $$y = \sin ^ { 2 } ( 4 x ) - \sin ^ { 10 } ( 4 x )$$ relative to axes taken along the bottom and left hand edge of the paper.
   The region of the poster below the curve is shaded and the region above the curve remains unshaded, as shown in Figure 2. Find the exact area of the poster which is shaded.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A particle \(P\) moves along a straight line. Initially \(P\) is at rest at the point \(O\) on the line. At time \(t\) seconds, where \(t \geqslant 0\)

• the displacement of \(P\) from \(O\) is \(x\) metres
• the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction
• the acceleration of \(P\) is \(\frac { 96 } { ( 3 t + 5 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction
  1. Show that, at time \(t\) seconds, \(v = p - \frac { q } { ( 3 t + 5 ) ^ { 2 } }\), where \(p\) and \(q\) are constants to be determined.
  2. Find the limiting value of \(v\) as \(t\) increases.
  3. Find the value of \(x\) when \(t = 2\)