1.08d Evaluate definite integrals: between limits

A small ball of mass \(m\) is projected vertically upwards from a point \(O\) with speed \(U\). The ball is subject to air resistance of magnitude \(mkv\), where \(v\) is the speed of the ball and \(k\) is a positive constant. Find, in terms of \(U\), \(g\) and \(k\), the maximum height above \(O\) reached by the ball. (8)

\includegraphics{figure_7} The diagram shows the curve with equation \(y = \sqrt{x}\), together with a set of \(n\) rectangles of unit width.

1. By considering the areas of these rectangles, explain why $$\sqrt{1} + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{n} > \int_0^n \sqrt{x} dx.$$ [2]
2. By drawing another set of rectangles and considering their areas, show that $$\sqrt{1} + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{n} < \int_1^{n+1} \sqrt{x} dx.$$ [3]
3. Hence find an approximation to \(\sum_{n=1}^{100} \sqrt{n}\), giving your answer correct to 2 significant figures. [3]

1. Solve the equation $$\sinh t + 7 \cosh t = 8,$$ expressing your answer in exact logarithmic form. [6]

A curve has equation \(y = \cosh 2x + 7 \sinh 2x\).

1. Using part (i), or otherwise, find, in an exact form, the coordinates of the points on the curve at which the gradient is 16. Show that there is no point on the curve at which the gradient is zero. Sketch the curve. [8]
2. Find, in an exact form, the positive value of \(a\) for which the area of the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = a\) is \(\frac{1}{2}\). [4]

$$f(x) = x - [x], \quad x \geq 0$$ where \([x]\) is the largest integer \(\leq x\). For example, \(f(3.7) = 3.7 - 3 = 0.7\); \(f(3) = 3 - 3 = 0\).

1. Sketch the graph of \(y = f(x)\) for \(0 \leq x < 4\). [3]
2. Find the value of \(p\) for which \(\int_2^p f(x) dx = 0.18\). [3]

Given that $$g(x) = \frac{1}{1+kx}, \quad x \geq 0, \quad k > 0,$$ and that \(x_0 = \frac{1}{2}\) is a root of the equation \(f(x) = g(x)\),

1. find the value of \(k\). [2]
2. Add a sketch of the graph of \(y = g(x)\) to your answer to part \((a)\). [1]

The root of \(f(x) = g(x)\) in the interval \(n < x < n + 1\) is \(x_n\), where \(n\) is an integer.

1. Prove that $$2 x_n^2 - (2n - 1)x_n - (n + 1) = 0.$$ [4]
2. Find the smallest value of \(n\) for which \(x_n - n < 0.05\). [4]

In this question you must show detailed reasoning. \includegraphics{figure_6} The diagram shows the curves \(y = \sqrt{2x + 9}\) and \(y = 4\mathrm{e}^{-2x} - 1\) which intersect on the \(y\)-axis. The shaded region is bounded by the curves and the \(x\)-axis. Determine the area of the shaded region, giving your answer in the form \(p + q \ln 2\) where \(p\) and \(q\) are constants to be determined. [8]

A mathematics department is designing a new emblem to place on the walls outside its classrooms. The design for the emblem is shown in the diagram below. \includegraphics{figure_5} The emblem is modelled by the region between the \(x\)-axis and the curve with parametric equations $$x = 1 + 0.2t - \cos t, \quad y = k \sin^2 t,$$ where \(k\) is a positive constant and \(0 \leq t \leq \pi\). Lengths are in metres and the area of the emblem must be \(1 \text{m}^2\).

1. Show that \(k \int_0^\pi (0.2 + \sin t - 0.2 \cos^2 t - \sin t \cos^2 t) dt = 1\). [3]
2. Determine the exact value of \(k\). [6]

A car, starting from rest, is driven along a horizontal track. The velocity of the car, \(v \text{m s}^{-1}\), at time \(t\) seconds, is modelled by the equation $$v = 0.48t^2 - 0.024t^3 \text{ for } 0 \leq t \leq 15$$

1. Find the distance the car travels during the first 10 seconds of its journey. [3 marks]
2. Find the maximum speed of the car. Give your answer to three significant figures. [4 marks]
3. Deduce the range of values of \(t\) for which the car is modelled as decelerating. [2 marks]

A particle, \(P\), is moving in a straight line with acceleration \(a\text{ m s}^{-2}\) at time \(t\) seconds, where $$a = 4 - 3t^2$$

1. Initially \(P\) is stationary. Find an expression for the velocity of \(P\) in terms of \(t\). [2 marks]
2. When \(t = 2\), the displacement of \(P\) from a fixed point, O, is 39 metres. Find the time at which \(P\) passes through O, giving your answer to three significant figures. Fully justify your answer. [5 marks]

Rakti makes open-topped cylindrical planters out of thin sheets of galvanised steel. She bends a rectangle of steel to make an open cylinder and welds the joint. She then welds this cylinder to the circumference of a circular base. \includegraphics{figure_11} The planter must have a capacity of \(8000\text{cm}^3\) Welding is time consuming, so Rakti wants the total length of weld to be a minimum. Calculate the radius, \(r\), and height, \(h\), of a planter which requires the minimum total length of weld. Fully justify your answers, giving them to an appropriate degree of accuracy. [9 marks]

A curve has equation \(y = 6x^2 + \frac{8}{x^2}\) and is sketched below for \(x > 0\) \includegraphics{figure_6} Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = a\) and \(x = 2a\), where \(a > 0\), giving your answer in terms of \(a\) [4 marks]

The graph of \(y = \frac{2x^3}{x^2 + 1}\) is shown for \(0 \leq x \leq 4\) Caroline is attempting to approximate the shaded area, A, under the curve using the trapezium rule by splitting the area into \(n\) trapezia.

1. When \(n = 4\)
   1. State the number of ordinates that Caroline uses. [1 mark]
   2. Calculate the area that Caroline should obtain using this method. Give your answer correct to two decimal places. [3 marks]
2. Show that the exact area of \(A\) is $$16 - \ln 17$$ Fully justify your answer. [5 marks]
3. Explain what would happen to Caroline's answer to part (a)(ii) as \(n \to \infty\) [1 mark]

Given that $$\int_0^{10} f(x) \, dx = 7$$ deduce the value of $$\int_0^{10} \left( f(x) + 1 \right) dx$$ Circle your answer. [1 mark] \(-3\) \quad \(7\) \quad \(8\) \quad \(17\)

\includegraphics{figure_5} Figure 5 shows a sketch of part of the curve \(y = 2x + \frac{8}{x^2} - 5\), \(x > 0\). The point \(A(4, \frac{7}{2})\) lies on C. The line \(l\) is the tangent to C at the point A. The region \(R\), shown shaded in figure 5 is bounded by the line \(l\), the curve C, the line with equation \(x = 1\) and the \(x\)-axis. Find the exact area of \(R\). (Solutions based entirely on graphical or numerical methods are not acceptable.)

In this question you must show detailed reasoning. The diagram shows part of the graph of \(y = x^3 - 4x\). \includegraphics{figure_3} Determine the total area enclosed by the curve and the \(x\)-axis. [6]

Show that \(\int_0^9 (3 + 4\sqrt{x})dx = 99\). [4]

Evaluate \(\int_0^{\frac{\pi}{12}} \cos 3x \, dx\), giving your answer in exact form. [3]

A curve has equation \(y = \cosh x\) Show that the arc length of the curve from \(x = a\) to \(x = b\), where \(0 < a < b\), is equal to $$\sinh b - \sinh a$$ [4 marks]

A parabola \(P_1\) has equation \(y^2 = 4ax\) where \(a > 0\) \(P_1\) is translated by the vector \(\begin{bmatrix} b \\ 0 \end{bmatrix}\), where \(b > 0\), to give the parabola \(P_2\)

1. The line \(y = mx\) is a tangent to \(P_2\) Prove that \(m = \pm\sqrt{\frac{a}{b}}\) Solutions using differentiation will be given no marks. [4 marks]
2. The line \(y = \sqrt{\frac{a}{b}} x\) meets \(P_2\) at the point \(D\). The finite region \(R\) is bounded by the \(x\)-axis, \(P_2\) and a line through \(D\) perpendicular to the \(x\)-axis. The region \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis to form a solid. Find, in terms of \(a\) and \(b\), the volume of this solid. Fully justify your answer. [5 marks]

The following algorithm produces a numerical approximation for the integral $$I = \int_A^B x^4 \, dx$$

Step 1	Start
Step 2	Input the values of A, B and N
Step 3	Let H = (B - A) / N
Step 4	Let C = H / 2
Step 5	Let D = 0
Step 6	Let D = D + A\(^4\) + B\(^4\)
Step 7	Let E = A
Step 8	Let E = E + H
Step 9	If E = B go to Step 12
Step 10	Let D = D + 2 × E\(^4\)
Step 11	Go to Step 8
Step 12	Let F = C × D
Step 13	Output F
Step 14	Stop

For the case when A = 1, B = 3 and N = 4,

1. 1. complete the table in the answer book to show the results obtained at each step of the algorithm.
   2. State the final output. [4]
2. Calculate, to 3 significant figures, the percentage error between the exact value of \(I\) and the value obtained from using the approximation to \(I\) in this case. [3]