1.08d Evaluate definite integrals: between limits

8

1. Find the quotient and remainder when \(8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7\) is divided by \(4 x ^ { 2 } + 1\).
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7 } { 4 x ^ { 2 } + 1 } \mathrm {~d} x\).

9 \includegraphics[max width=\textwidth, alt={}, center]{98001cfe-46a1-4c8f-9230-c140ebff6176-14_535_1082_274_520} The diagram shows part of the curve \(y = ( 3 - x ) \mathrm { e } ^ { - \frac { 1 } { 3 } x }\) for \(x \geqslant 0\), and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the area of the shaded region bounded by the curve and the axes, giving your answer in terms of e.

9

1. Find the quotient and remainder when \(x ^ { 4 } + 16\) is divided by \(x ^ { 2 } + 4\).
2. Hence show that \(\int _ { 2 } ^ { 2 \sqrt { 3 } } \frac { x ^ { 4 } + 16 } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 4 } { 3 } ( \pi + 4 )\).

7 A particle moves in a straight line through the point \(O\). The displacement of the particle from \(O\) at time \(t \mathrm {~s}\) is \(s \mathrm {~m}\), where $$\begin{array} { l l } s = t ^ { 2 } - 3 t + 2 & \text { for } 0 \leqslant t \leqslant 6 , \\ s = \frac { 24 } { t } - \frac { t ^ { 2 } } { 4 } + 25 & \text { for } t \geqslant 6 . \end{array}$$

1. Find the value of \(t\) when the particle is instantaneously at rest during the first 6 seconds of its motion.
   At \(t = 6\), the particle hits a barrier at a point \(P\) and rebounds.
2. Find the velocity with which the particle arrives at \(P\) and also the velocity with which the particle leaves \(P\).
3. Find the total distance travelled by the particle in the first 10 seconds of its motion.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A particle moves in a straight line. It starts from rest from a fixed point \(O\) on the line. Its velocity at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t ^ { 2 } - 8 t ^ { \frac { 3 } { 2 } } + 10 t\).

1. Find the displacement of the particle from \(O\) when \(t = 1\).
2. Show that the minimum velocity of the particle is \(- 125 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 A cyclist starts from rest at a fixed point \(O\) and moves in a straight line, before coming to rest \(k\) seconds later. The acceleration of the cyclist at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 2 t ^ { - \frac { 1 } { 2 } } - \frac { 3 } { 5 } t ^ { \frac { 1 } { 2 } }\) for \(0 < t \leqslant k\).

1. Find the value of \(k\).
2. Find the maximum speed of the cyclist.
3. Find an expression for the displacement from \(O\) in terms of \(t\). Hence find the total distance travelled by the cyclist from the time at which she reaches her maximum speed until she comes to rest.

7 A particle \(P\) moves in a straight line, starting from a point \(O\) with velocity \(1.72 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle, \(t \mathrm {~s}\) after leaving \(O\), is given by \(a = 0.1 t ^ { \frac { 3 } { 2 } }\).

1. Find the value of \(t\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the displacement of \(P\) from \(O\) when \(t = 2\), giving your answer correct to 2 decimal places. [3]

5 A particle \(P\) moves in a straight line. It starts at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 4 t ^ { 2 } - 20 t + 21\).

1. Find the values of \(t\) for which \(P\) is at instantaneous rest.
2. Find the initial acceleration of \(P\).
3. Find the minimum velocity of \(P\).
4. Find the distance travelled by \(P\) during the time when its velocity is negative.

6 A particle \(P\) moves in a straight line starting from a point \(O\) and comes to rest 14 s later. At time \(t \mathrm {~s}\) after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by $$\begin{array} { l l } v = p t ^ { 2 } - q t & 0 \leqslant t \leqslant 6 \\ v = 63 - 4.5 t & 6 \leqslant t \leqslant 14 \end{array}$$ where \(p\) and \(q\) are positive constants.
The acceleration of \(P\) is zero when \(t = 2\).

1. Given that there are no instantaneous changes in velocity, find \(p\) and \(q\).
2. Sketch the velocity-time graph.
3. Find the total distance travelled by \(P\) during the 14 s . \includegraphics[max width=\textwidth, alt={}, center]{e1b91e54-a3ae-436c-a4f7-7095891f7034-10_326_1109_255_520} Two particles \(A\) and \(B\) of masses 2 kg and 3 kg respectively are connected by a light inextensible string. Particle \(B\) is on a smooth fixed plane which is at an angle of \(18 ^ { \circ }\) to horizontal ground. The string passes over a fixed smooth pulley at the top of the plane. Particle \(A\) hangs vertically below the pulley and is 0.45 m above the ground (see diagram). The system is released from rest with the string taut. When \(A\) reaches the ground, the string breaks. Find the total distance travelled by \(B\) before coming to instantaneous rest. You may assume that \(B\) does not reach the pulley.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 A cyclist starts from rest at a point \(A\) and travels along a straight road \(A B\), coming to rest at \(B\). The displacement of the cyclist from \(A\) at time \(t \mathrm {~s}\) after the start is \(s \mathrm {~m}\), where $$s = 0.004 \left( 75 t ^ { 2 } - t ^ { 3 } \right)$$

1. Show that the distance \(A B\) is 250 m .
2. Find the maximum velocity of the cyclist.

5 A particle \(P\) moves in a straight line, starting from rest at a point \(O\) on the line. At time \(t \mathrm {~s}\) after leaving \(O\) the acceleration of \(P\) is \(k \left( 16 - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 2 }\), where \(k\) is a positive constant, and the displacement from \(O\) is \(s \mathrm {~m}\). The velocity of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 4\).

1. Show that \(s = \frac { 1 } { 64 } t ^ { 2 } \left( 96 - t ^ { 2 } \right)\).
2. Find the speed of \(P\) at the instant that it returns to \(O\).
3. Find the maximum displacement of the particle from \(O\).

5 A particle \(P\) moves in a straight line that passes through the origin \(O\). The velocity of \(P\) at time \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 20 t - t ^ { 3 }\). At time \(t = 0\) the particle is at rest at a point whose displacement from \(O\) is - 36 m .

1. Find an expression for the displacement of \(P\) from \(O\) in terms of \(t\).
2. Find the displacement of \(P\) from \(O\) when \(t = 4\).
3. Find the values of \(t\) for which the particle is at \(O\).

2 A motorcyclist starts from rest at \(A\) and travels in a straight line until he comes to rest again at \(B\). The velocity of the motorcyclist \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t - 0.01 t ^ { 2 }\). Find

1. the time taken for the motorcyclist to travel from \(A\) to \(B\),
2. the distance \(A B\).

7 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-4_719_1059_264_543} An object \(P\) travels from \(A\) to \(B\) in a time of 80 s . The diagram shows the graph of \(v\) against \(t\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(A\). The graph consists of straight line segments for the intervals \(0 \leqslant t \leqslant 10\) and \(30 \leqslant t \leqslant 80\), and a curved section whose equation is \(v = - 0.01 t ^ { 2 } + 0.5 t - 1\) for \(10 \leqslant t \leqslant 30\). Find

1. the maximum velocity of \(P\),
2. the distance \(A B\).

6 A particle travels in a straight line from a point \(P\) to a point \(Q\). Its velocity \(t\) seconds after leaving \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 4 t - \frac { 1 } { 16 } t ^ { 3 }\). The distance \(P Q\) is 64 m .

1. Find the time taken for the particle to travel from \(P\) to \(Q\).
2. Find the set of values of \(t\) for which the acceleration of the particle is positive.

3 A particle \(P\) travels from a point \(O\) along a straight line and comes to instantaneous rest at a point \(A\). The velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.027 \left( 10 t ^ { 2 } - t ^ { 3 } \right)\). Find

1. the distance \(O A\),
2. the maximum velocity of \(P\) while moving from \(O\) to \(A\).

7 A car driver makes a journey in a straight line from \(A\) to \(B\), starting from rest. The speed of the car increases to a maximum, then decreases until the car is at rest at \(B\). The distance travelled by the car \(t\) seconds after leaving \(A\) is \(0.0000117 \left( 400 t ^ { 3 } - 3 t ^ { 4 } \right)\) metres.

1. Find the distance \(A B\).
2. Find the maximum speed of the car.
3. Find the acceleration of the car
   1. as it starts from \(A\),
   2. as it arrives at \(B\).
   3. Sketch the velocity-time graph for the journey.

4 An aeroplane moves along a straight horizontal runway before taking off. It starts from rest at \(O\) and has speed \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the instant it takes off. While the aeroplane is on the runway at time \(t\) seconds after leaving \(O\), its acceleration is \(( 1.5 + 0.012 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find

1. the value of \(t\) at the instant the aeroplane takes off,
2. the distance travelled by the aeroplane on the runway.

1

1. Find the Maclaurin series for \(\sin ^ { - 1 } x\) up to and including the term in \(x ^ { 3 }\).
2. Deduce an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 5 } } \frac { 1 } { \sqrt { 1 - u ^ { 2 } } } \mathrm {~d} u\), giving your answer as a fraction.

7 A particle starts from rest at the point \(A\) and travels in a straight line until it reaches the point \(B\). The velocity of the particle \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.009 t ^ { 2 } - 0.0001 t ^ { 3 }\). Given that the velocity of the particle when it reaches \(B\) is zero, find

1. the time taken for the particle to travel from \(A\) to \(B\),
2. the distance \(A B\),
3. the maximum velocity of the particle.

6 A particle \(P\) starts from rest at \(O\) and travels in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by \(v = 8 t - 2 t ^ { 2 }\) for \(0 \leqslant t \leqslant 3\), and \(v = \frac { 54 } { t ^ { 2 } }\) for \(t > 3\). Find

1. the distance travelled by \(P\) in the first 3 seconds,
2. an expression in terms of \(t\) for the displacement of \(P\) from \(O\), valid for \(t > 3\),
3. the value of \(v\) when the displacement of \(P\) from \(O\) is 27 m .

4 The velocity of a particle \(t \mathrm {~s}\) after it starts from rest is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 1.25 t - 0.05 t ^ { 2 }\). Find

1. the initial acceleration of the particle,
2. the displacement of the particle from its starting point at the instant when its acceleration is \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

3 A particle \(P\) moves along a straight line for 100 s . It starts at a point \(O\) and at time \(t\) seconds after leaving \(O\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.00004 t ^ { 3 } - 0.006 t ^ { 2 } + 0.288 t$$

1. Find the values of \(t\) at which the acceleration of \(P\) is zero.
2. Find the displacement of \(P\) from \(O\) when \(t = 100\).

6 A particle \(P\) starts from rest at a point \(O\) of a straight line and moves along the line. The displacement of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(x \mathrm {~m}\), where $$x = 0.08 t ^ { 2 } - 0.0002 t ^ { 3 }$$

1. Find the value of \(t\) when \(P\) returns to \(O\) and find the speed of \(P\) as it passes through \(O\) on its return.
2. For the motion of \(P\) until the instant it returns to \(O\), find
   1. the total distance travelled,
   2. the average speed.

7 A racing car is moving in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t \mathrm {~s}\) after the car starts from rest is given by $$\begin{array} { l l } a = 15 t - 3 t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 5 \\ a = - \frac { 625 } { t ^ { 2 } } & \text { for } 5 < t \leqslant k \end{array}$$ where \(k\) is a constant.

1. Find the maximum acceleration of the car in the first five seconds of its motion.
2. Find the distance of the car from its starting point when \(t = 5\).
3. The car comes to rest when \(t = k\). Find the value of \(k\).