1.08g Integration as limit of sum: Riemann sums

4.(a)Use the trapezium rule with 4 strips to find an approximate value for $$\int _ { 0 } ^ { 1 } 16 ^ { x } d x$$ (b)Use the trapezium rule with \(n\) strips to write down an expression that would give an approximate value for $$\int _ { 0 } ^ { 1 } 16 ^ { x } d x$$ (c)Hence show that $$\int _ { 0 } ^ { 1 } 16 ^ { x } \mathrm {~d} x = \lim _ { n \rightarrow \infty } \left( \frac { 1 } { n } \left( 1 + 16 ^ { \frac { 1 } { n } } + \ldots + 16 ^ { \frac { n - 1 } { n } } \right) \right)$$ (d)Use integration to determine the exact value of $$\int _ { 0 } ^ { 1 } 16 ^ { x } d x$$ Given that the limit exists,
(e)use part(c)and the answer to part(d)to determine the exact value of $$\lim _ { x \rightarrow 0 } \frac { 16 ^ { x } - 1 } { x }$$

3 The diagram shows the curve \(y = \frac { 1 } { x ^ { 3 } }\) for \(1 \leqslant x \leqslant n\) where \(n\) is an integer. A set of ( \(n - 1\) ) rectangles of unit width is drawn under the curve. \includegraphics[max width=\textwidth, alt={}, center]{736932f1-4007-4a04-a08b-2551db0b136c-2_611_947_1103_557}

1. Write down the sum of the areas of the rectangles.
2. Hence show that \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 3 } } < \frac { 3 } { 2 }\).

6 Alex is investigating the area, \(A\), under the graph of \(y = x ^ { 2 }\) between \(x = 1\) and \(x = 1.5\). They draw the graph, together with rectangles of width \(\delta x = 0.1\), and varying heights \(y\). \includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-06_531_714_356_251}

1. Use the rectangles in the diagram to show that lower and upper bounds for the area \(A\) are 0.73 and 0.855 respectively.
2. Alex finds lower and upper bounds for the area \(A\), using widths \(\delta x\) of decreasing size. The results are shown in the table. Where relevant, values are given correct to 3 significant figures.

   Width \(\delta x\)	0.1	0.05	0.025	0.0125
   Lower bound for area \(A\)	0.73	0.761	0.776	0.784
   Upper bound for area \(A\)	0.855	0.823	0.807	0.799

   Use Alex's results to estimate the value of \(A\) correct to \(\mathbf { 2 }\) significant figures. Give a brief justification for your estimate.
3. Write down an expression, in terms of \(y\) and \(\delta x\), for the exact value of the area \(A\).

4. a. Express \(\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x \quad\) as an integral.
b. Hence show that $$\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x = \ln k$$ where \(k\) is a constant to be found.

1. (a) Express \(\lim _ { \delta x \rightarrow 0 } \sum _ { x = 2.1 } ^ { 6.3 } \frac { 2 } { x } \delta x\) as an integral.
   (b) Hence show that

$$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 2.1 } ^ { 6.3 } \frac { 2 } { x } \delta x = \ln k$$ where \(k\) is a constant to be found.

1. The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground.

The results are given in the table below with the time in seconds and the speed in \(\mathrm { ms } ^ { - 1 }\). Using all of this information,

1. estimate the length of runway used by the jet to take off. Given that the jet accelerated smoothly in these 25 seconds,
2. explain whether your answer to part (a) is an underestimate or an overestimate of the length of runway used by the jet to take off.

5. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve with equation \(y = \sqrt { x }\) The point \(P ( x , y )\) lies on the curve.
The rectangle, shown shaded on Figure 3, has height \(y\) and width \(\delta x\).
Calculate $$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 4 } ^ { 9 } \sqrt { x } \delta x$$

7 The diagram shows part of the curve \(y = x ^ { 2 }\) for \(0 \leqslant x \leqslant p\), where \(p\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_736_543_669_762} The area \(A\) of the region enclosed by the curve, the \(x\)-axis and the line \(x = p\) is given approximately by the sum \(S\) of the areas of \(n\) rectangles, each of width \(h\), where \(h\) is small and \(n h = p\). The first three such rectangles are shown in the diagram.

1. Find an expression for \(S\) in terms of \(n\) and \(h\).
2. Use the identity \(\sum _ { r = 1 } ^ { n } r ^ { 2 } \equiv \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to show that \(S = \frac { 1 } { 6 } p ( p + h ) ( 2 p + h )\).
3. Show how to use this result to find \(A\) in terms of \(p\).

7 \includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-04_673_1285_1176_429} The diagram shows the curve with equation \(y = \sqrt { x }\). A set of \(N\) rectangles of unit width is drawn, starting at \(x = 1\) and ending at \(x = N + 1\), where \(N\) is an integer (see diagram).

1. By considering the areas of these rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } < \int _ { 1 } ^ { N + 1 } \sqrt { x } \mathrm {~d} x$$
2. By considering the areas of another set of rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } > \int _ { 0 } ^ { N } \sqrt { x } \mathrm {~d} x$$
3. Hence find, in terms of \(N\), limits between which \(\sum _ { r = 1 } ^ { N } \sqrt { r }\) lies. \section*{Jan 2006}

14

1. Express \(\lim _ { \delta y \rightarrow 0 } \sum _ { 0 } ^ { h } \left( h ^ { 2 } - y ^ { 2 } \right) \delta y\) as an integral.
2. Hence show that \(V = \frac { 2 } { 3 } \pi r ^ { 2 } h\), as given in line 41 .

\includegraphics{figure_4} The diagram shows the curve with equation \(y = \ln x\) for \(x \geqslant 1\), together with a set of \((N-1)\) rectangles of unit width.

1. By considering the sum of the areas of these rectangles, show that $$\ln N! > N \ln N - N + 1.$$ [5]
2. Use a similar method to find, in terms of \(N\), an upper bound for \(\ln N!\). [3]

\includegraphics{figure_3} The diagram shows the curve with equation \(y = 1 - x^2\) for \(0 \leq x \leq 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).

1. By considering the sum of the areas of the rectangles, show that $$\int_0^1 (1 - x^2) \, dx < \frac{4n^2 + 3n - 1}{6n^2}.$$ [4]
2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int_0^1 (1 - x^2) \, dx\). [4]

1. State the sum of the series \(1 + z + z^2 + \ldots + z^{n-1}\), for \(z \neq 1\). [1]
2. By letting \(z = \cos\theta + i\sin\theta\), where \(\cos\theta \neq 1\), show that $$1 + \cos\theta + \cos 2\theta + \ldots + \cos(n-1)\theta = \frac{1}{2}\left(1 - \cos n\theta + \frac{\sin n\theta \sin\theta}{1 - \cos\theta}\right).$$ [7]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = \cos x\) for \(0 \leq x \leq 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).

1. By considering the sum of the areas of these rectangles, show that $$\int_0^1 \cos x\,dx < \frac{1}{2n}\left(1 - \cos 1 + \frac{\sin 1\sin\frac{1}{n}}{1 - \cos\frac{1}{n}}\right).$$ [4]
2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int_0^1 \cos x\,dx\). [3]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = e^{1-x}\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).

1. By considering the sum of the areas of these rectangles, show that \(\int_0^1 e^{1-x} \, dx < U_n\), where $$U_n = \frac{e-1}{n(1-e^{-1})}.$$ [4]
2. Use a similar method to find, in terms of \(n\), a lower bound \(L_n\) for \(\int_0^1 e^{1-x} \, dx\). [4]
3. Show that \(\lim_{n \to \infty}(U_n - L_n) = 0\). [2]
4. Use the Maclaurin's series for \(e^x\) given in the list of formulae (MF19) to find the first three terms of the series expansion of \(z(1-e^{-z})\), in ascending powers of \(z\), and deduce the value of \(\lim_{n \to \infty}(U_n)\). [3]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = \frac{1}{x+1}\). A set of \(n\) rectangles of unit width is drawn, starting at \(x = 0\) and ending at \(x = n\), where \(n\) is an integer.

1. By considering the areas of these rectangles, explain why $$\frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n+1} < \ln(n+1).$$ [5]
2. By considering the areas of another set of rectangles, show that $$1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} > \ln(n+1).$$ [2]
3. Hence show that $$\ln(n+1) + \frac{1}{n+1} < \sum_{r=1}^{n+1} \frac{1}{r} < \ln(n+1) + 1.$$ [2]
4. State, with a reason, whether \(\sum_{r=1}^{\infty} \frac{1}{r}\) is convergent. [2]

\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]

Fig. 14.1 shows the curve with equation \(y = \frac{1}{1 + x^2}\), together with 5 rectangles of equal width. \includegraphics{figure_14_1} Fig. 14.2 shows the coordinates of the points A, B, C, D, E and F. \includegraphics{figure_14_2}

1. Use the 5 rectangles shown in Fig. 14.1 and the information in Fig. 14.2 to show that a lower bound for \(\int_0^1 \frac{1}{1 + x^2}\,dx\) is 0.7337, correct to 4 decimal places. [2]
2. Use the 5 rectangles shown in Fig. 14.1 and the information in Fig. 14.2 to calculate an upper bound for \(\int_0^1 \frac{1}{1 + x^2}\,dx\) correct to 4 decimal places. [2]
3. Hence find the length of the interval in which your answers to parts (a) and (b) indicate the value of \(\int_0^1 \frac{1}{1 + x^2}\,dx\) lies. [1]

Amit uses \(n\) rectangles, each of width \(\frac{1}{n}\), to calculate upper and lower bounds for \(\int_0^1 \frac{1}{1 + x^2}\,dx\), using different values of \(n\). His results are shown in Fig. 14.3. \includegraphics{figure_14_3}

1. Find the length of the smallest interval in which Amit now knows \(\int_0^1 \frac{1}{1 + x^2}\,dx\) lies. [2]
2. Without doing any calculation, explain how Amit could find a smaller interval which contains the value of \(\int_0^1 \frac{1}{1 + x^2}\,dx\). [1]