1.08d Evaluate definite integrals: between limits

A particle moves in a straight line. It starts from rest at a fixed point \(O\) on the line. Its acceleration at time \(t\) s after leaving \(O\) is \(a\) m s\(^{-2}\), where \(a = 0.4t^3 - 4.8t^2\).

1. Show that, in the subsequent motion, the acceleration of the particle when it comes to instantaneous rest is \(16\) m s\(^{-2}\). [6]
2. Find the displacement of the particle from \(O\) at \(t = 5\). [3]

A particle starts from rest and moves in a straight line. The velocity of the particle at time \(t\) s after the start is \(v\) m s\(^{-1}\), where $$v = -0.01t^3 + 0.22t^2 - 0.4t.$$

1. Find the two positive values of \(t\) for which the particle is instantaneously at rest. [2]
2. Find the time at which the acceleration of the particle is greatest. [3]
3. Find the distance travelled by the particle while its velocity is positive. [4]

A particle moves in a straight line, starting from rest at a point \(O\), and comes to instantaneous rest at a point \(P\). The velocity of the particle at time \(t\) s after leaving \(O\) is \(v\) m s\(^{-1}\), where $$v = 0.6t^2 - 0.12t^3.$$

1. Show that the distance \(OP\) is 6.25 m. [5]

On another occasion, the particle also moves in the same straight line. On this occasion, the displacement of the particle at time \(t\) s after leaving \(O\) is \(s\) m, where $$s = kt^3 + ct^5.$$ It is given that the particle passes point \(P\) with velocity 1.25 m s\(^{-1}\) at time \(t = 5\).

1. Find the values of the constants \(k\) and \(c\). [5]

1. Find the acceleration of the particle at time \(t = 5\). [2]

A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in m s\(^{-1}\), at time \(t\) s after leaving \(O\) is given by $$v = 0.6t - 0.03t^2.$$

1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time. [4]
2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity. [6]

$$f(x) = Ax^3 + 6x^2 - 4x + B$$ where \(A\) and \(B\) are constants. Given that

• \((x + 2)\) is a factor of \(f(x)\)
• \(\int_{-3}^{5} f(x)dx = 176\)

Find the value of \(A\) and the value of \(B\). [7]

\includegraphics{figure_2} Figure 2 shows a sketch of the curve defined by the parametric equations $$x = t^2 + 2t \quad y = \frac{2}{t(3-t)} \quad a \leq t \leq b$$ where \(a\) and \(b\) are constants. The ends of the curve lie on the line with equation \(y = 1\)

1. Find the value of \(a\) and the value of \(b\) [2]

The region \(R\), shown shaded in Figure 2, is bounded by the curve and the line with equation \(y = 1\)

1. Show that the area of region \(R\) is given by $$M - k \int_a^b \frac{t+1}{t(3-t)} dt$$ where \(M\) and \(k\) are constants to be found. [5]
2. 1. Write \(\frac{t+1}{t(3-t)}\) in partial fractions.
   2. Use algebraic integration to find the exact area of \(R\), giving your answer in simplest form. [6]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \includegraphics{figure_2} Figure 2 shows a sketch of part of the curve with equation $$y = \frac{12\sqrt{x}}{(2x^2 + 3)^3}$$ The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = \frac{1}{\sqrt{2}}\), the \(x\)-axis and the line with equation \(x = k\). This region is rotated through \(360°\) about the \(x\)-axis to form a solid of revolution. Given that the volume of this solid is \(\frac{713\pi}{648}\), use algebraic integration to find the exact value of the constant \(k\). [6]

\includegraphics{figure_2} Part of the design of a stained glass window is shown in Fig. 2. The two loops enclose an area of blue glass. The remaining area within the rectangle \(ABCD\) is red glass. The loops are described by the curve with parametric equations $$x = 3 \cos t, \quad y = 9 \sin 2t, \quad 0 \leq t < 2\pi.$$

1. Find the cartesian equation of the curve in the form \(y^2 = f(x)\). [4]
2. Show that the shaded area in Fig. 2, enclosed by the curve and the \(x\)-axis, is given by $$\int_0^{\frac{\pi}{2}} A \sin 2t \sin t \, dt, \text{ stating the value of the constant } A.$$ [3]
3. Find the value of this integral. [4]

The sides of the rectangle \(ABCD\), in Fig. 2, are the tangents to the curve that are parallel to the coordinate axes. Given that 1 unit on each axis represents 1 cm,

1. find the total area of the red glass. [4]

\includegraphics{figure_1} A table top, in the shape of a parallelogram, is made from two types of wood. The design is shown in Fig. 1. The area inside the ellipse is made from one type of wood, and the surrounding area is made from a second type of wood. The ellipse has parametric equations, $$x = 5 \cos \theta, \quad y = 4 \sin \theta, \quad 0 \leq \theta < 2\pi$$ The parallelogram consists of four line segments, which are tangents to the ellipse at the points where \(\theta = \alpha\), \(\theta = -\alpha\), \(\theta = \pi - \alpha\), \(\theta = -\pi + \alpha\).

1. Find an equation of the tangent to the ellipse at \((5 \cos \alpha, 4 \sin \alpha)\), and show that it can be written in the form $$5y \sin \alpha + 4x \cos \alpha = 20.$$ [4]
2. Find by integration the area enclosed by the ellipse. [4]
3. Hence show that the area enclosed between the ellipse and the parallelogram is $$\frac{80}{\sin 2\alpha} - 20\pi.$$ [4]
4. Given that \(0 < \alpha < \frac{\pi}{4}\), find the value of \(\alpha\) for which the areas of two types of wood are equal. [3]

\includegraphics{figure_12} Figure 1 shows the cross-section \(R\) of an artificial ski slope. The slope is modelled by the curve with equation $$y = \frac{10}{\sqrt{4x^2 + 9}}, \quad 0 \leq x \leq 5.$$ Given that 1 unit on each axis represents 10 metres, use integration to calculate the area \(R\). Show your method clearly and give your answer to 2 significant figures. [7]

\includegraphics{figure_13} A rope is hung from points \(P\) and \(Q\) on the same horizontal level, as shown in Fig. 2. The curve formed by the rope is modelled by the equation $$y = a \cosh\left(\frac{x}{a}\right), \quad -ka \leq x \leq ka,$$ where \(a\) and \(k\) are positive constants.

1. Prove that the length of the rope is \(2a \sinh k\). [5]

Given that the length of the rope is \(8a\),

1. find the coordinates of \(Q\), leaving your answer in terms of natural logarithms and surds, where appropriate. [4]

\includegraphics{figure_1} The parametric equations of the curve \(C\) shown in Figure 1 are $$x = a(t - \sin t), \quad y = a(1 - \cos t), \quad 0 \leq t \leq 2\pi$$ Find, by using integration, the length of \(C\). (Total 6 marks)

A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\) m s\(^{-1}\) in the direction of \(x\) increasing, where $$v = (t - 2)(3t - 10), \quad t \geq 0$$ When \(t = 0\), \(P\) is at the origin \(O\).

1. Find the acceleration of \(P\) when \(t = 3\) [3]
2. Find the total distance travelled by \(P\) in the first 3 seconds of its motion. [6]
3. Show that \(P\) never returns to \(O\). [2]

At time \(t\) seconds, \(t \geq 0\), a particle \(P\) has velocity \(\mathbf{v}\) m s\(^{-1}\), where $$\mathbf{v} = (27 - 3t^2)\mathbf{i} + (8 - t^3)\mathbf{j}$$ When \(t = 1\), the particle \(P\) is at the point with position vector \(\mathbf{r}\) m relative to a fixed origin \(O\), where \(\mathbf{r} = -5\mathbf{i} + 2\mathbf{j}\) Find

1. the magnitude of the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(\mathbf{i}\), [5]
2. the position vector of \(P\) at the instant when \(t = 3\) [5]

\includegraphics{figure_2} At time \(t = 0\) a small package is projected from a point \(B\) which is \(2.4\) m above a point \(A\) on horizontal ground. The package is projected with speed \(23.75\) m s\(^{-1}\) at an angle \(α\) to the horizontal, where \(\tan α = \frac{4}{5}\). The package strikes the ground at point \(C\), as shown in Fig. 2. The package is modelled as a particle moving freely under gravity.

1. Find the time taken for the package to reach \(C\). [5]

A lorry moves along the line \(AC\), approaching \(A\) with constant speed 18 m s\(^{-1}\). At time \(t = 0\) the rear of the lorry passes \(A\) and the lorry starts to slow down. It comes to rest 7 seconds later. The acceleration, \(a\) m s\(^{-2}\) of the lorry at time \(t\) seconds is given by $$a = -\frac{1}{4}t^2, \quad 0 \leq t \leq 7.$$

1. Find the speed of the lorry at time \(t\) seconds. [3]

1. Hence show that \(T = 6\). [3]

1. Show that when the package reaches \(C\) it is just under 10 m behind the rear of the moving lorry. [5]

END

A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \((4t - 8)\) m s\(^{-2}\), measured in the direction of \(x\) increasing. The velocity of \(P\) at time \(t\) seconds is \(v\) m s\(^{-1}\). Given that \(v = 6\) when \(t = 0\), find

1. \(v\) in terms of \(t\), [4]
2. the distance between the two points where \(P\) is instantaneously at rest. [7]

A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \text{ ms}^{-1}\) in the positive \(x\)-direction, where \(v = 3t^2 - 4t + 3\). When \(t = 0\), \(P\) is at the origin \(O\). Find the distance of \(P\) from \(O\) when \(P\) is moving with minimum velocity. [8]

A particle \(P\) is moving in a plane. At time \(t\) seconds, \(P\) is moving with velocity \(\mathbf{v}\) m s\(^{-1}\), where \(\mathbf{v} = 2t\mathbf{i} - 3t^2\mathbf{j}\). Find

1. the speed of \(P\) when \(t = 4\) [2]
2. the acceleration of \(P\) when \(t = 4\) [3]

Given that \(P\) is at the point with position vector \((-4\mathbf{i} + \mathbf{j})\) m when \(t = 1\),

1. find the position vector of \(P\) when \(t = 4\) [5]

A particle \(P\) of mass 0.25 kg moves under the action of a single force \(\mathbf{F}\) newtons. At time \(t\) seconds, the velocity of \(P\) is \(\mathbf{v}\) m s\(^{-1}\), where $$\mathbf{v} = (2 - 4t)\mathbf{i} + (t^2 + 2t)\mathbf{j}$$ When \(t = 0\), \(P\) is at the point with position vector \((2\mathbf{i} - 4\mathbf{j})\) m with respect to a fixed origin \(O\). When \(t = 3\), \(P\) is at the point \(A\). Find

1. the momentum of \(P\) when \(t = 3\), [2]
2. the magnitude of \(\mathbf{F}\) when \(t = 3\), [6]
3. the position vector of \(A\). [5]

A particle moving in a straight line starts from rest at the point \(O\) at time \(t = 0\). At time \(t\) seconds, the velocity \(v\) m s\(^{-1}\) of the particle is given by $$v = 3t(t - 4), \quad 0 \leq t \leq 5,$$ $$v = 75t^{-1}, \quad 5 \leq t \leq 10.$$

1. Sketch a velocity-time graph for the particle for \(0 \leq t \leq 10\). [3]
2. Find the set of values of \(t\) for which the acceleration of the particle is positive. [2]
3. Show that the total distance travelled by the particle in the interval \(0 \leq t \leq 5\) is \(39\) m. [3]
4. Find, to \(3\) significant figures, the value of \(t\) at which the particle returns to \(O\). [5]

Given that $$\frac{dy}{dx} = 3\sqrt{x} - x^2,$$ and that \(y = \frac{2}{3}\) when \(x = 1\), find the value of \(y\) when \(x = 4\). [7]