Bounds using rectangles

3 \includegraphics[max width=\textwidth, alt={}, center]{fa2213b3-480c-44cb-8ba0-ebd2b94d3d90-04_851_805_251_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 3 }\) 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 \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x < U _ { n }\), where $$\mathrm { U } _ { \mathrm { n } } = \left( \frac { \mathrm { n } + 1 } { 2 \mathrm { n } } \right) ^ { 2 }$$
2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x\).
3. Find the least value of \(n\) such that \(\mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } } < 10 ^ { - 3 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{fd247a71-4680-45d8-89d2-fef17ed3a5e9-04_851_805_251_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 3 }\) 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 \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x < U _ { n }\), where $$\mathrm { U } _ { \mathrm { n } } = \left( \frac { \mathrm { n } + 1 } { 2 \mathrm { n } } \right) ^ { 2 }$$
2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x\).
3. Find the least value of \(n\) such that \(\mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } } < 10 ^ { - 3 }\).

4 The diagram shows the curve with equation \(\mathrm { y } = 2 ^ { \mathrm { x } }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(N\) rectangles each of width \(\frac { 1 } { N }\). \includegraphics[max width=\textwidth, alt={}, center]{114ece0d-558d-4c02-8a77-034b3681cff9-06_824_1161_376_450}

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } 2 ^ { x } d x < U _ { N }\), where $$\mathrm { U } _ { \mathrm { N } } = \frac { 2 ^ { \frac { 1 } { \mathrm {~N} } } } { \mathrm {~N} \left( 2 ^ { \frac { 1 } { \mathrm {~N} } } - 1 \right) }$$
2. Use a similar method to find, in terms of \(N\), a lower bound \(\mathrm { L } _ { \mathrm { N } }\) for \(\int _ { 0 } ^ { 1 } 2 ^ { x } \mathrm {~d} x\).
3. Find the least value of \(N\) such that \(\mathrm { U } _ { \mathrm { N } } - \mathrm { L } _ { \mathrm { N } } < 10 ^ { - 4 }\).

4 The diagram shows the curve with equation \(\mathrm { y } = 2 ^ { \mathrm { x } }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(N\) rectangles each of width \(\frac { 1 } { N }\). \includegraphics[max width=\textwidth, alt={}, center]{69c540e1-1dad-45a1-9809-7629d16260e0-06_824_1161_376_450}

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } 2 ^ { x } d x < U _ { N }\), where $$\mathrm { U } _ { \mathrm { N } } = \frac { 2 ^ { \frac { 1 } { \mathrm {~N} } } } { \mathrm {~N} \left( 2 ^ { \frac { 1 } { \mathrm {~N} } } - 1 \right) }$$
2. Use a similar method to find, in terms of \(N\), a lower bound \(\mathrm { L } _ { \mathrm { N } }\) for \(\int _ { 0 } ^ { 1 } 2 ^ { x } \mathrm {~d} x\).
3. Find the least value of \(N\) such that \(\mathrm { U } _ { \mathrm { N } } - \mathrm { L } _ { \mathrm { N } } < 10 ^ { - 4 }\).

6 \includegraphics[max width=\textwidth, alt={}, center]{23b06b1c-997f-425d-ae3d-bd4cc1295605-10_771_1146_260_497} The diagram shows the curve with equation \(\mathrm { y } = \ln ( 1 + \mathrm { x } )\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles each of width \(\frac { 1 } { n }\).

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } \ln ( 1 + x ) d x < U _ { n }\), where $$U _ { n } = \frac { 1 } { n } \ln \frac { ( 2 n ) ! } { n ! } - \ln n$$
2. Use a similar method to find, in terms of \(n\), a lower bound \(\mathrm { L } _ { \mathrm { n } }\) for \(\int _ { 0 } ^ { 1 } \ln ( 1 + x ) \mathrm { d } x\).
3. By simplifying \(\mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } }\), show that \(\lim _ { \mathrm { n } \rightarrow \infty } \left( \mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } } \right) = 0\).

6 \includegraphics[max width=\textwidth, alt={}, center]{d421652f-576d-4843-abbf-54404e225fec-10_1015_988_260_577} The diagram shows the curve with equation \(\mathrm { y } = ( 1 - \mathrm { x } ) ^ { 2 }\) 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 \(\int _ { 0 } ^ { 1 } ( 1 - x ) ^ { 2 } d x < U _ { n }\), where $$U _ { n } = \frac { 2 n ^ { 2 } + 3 n + 1 } { 6 n ^ { 2 } }$$
2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } ( 1 - x ) ^ { 2 } d x\).
3. Show that \(\lim _ { n \rightarrow \infty } \left( U _ { n } - L _ { n } \right) = 0\).

5 \includegraphics[max width=\textwidth, alt={}, center]{bca7281b-a6a9-4b4c-94e5-3da2a561ad86-08_663_1152_260_452} The diagram shows the curve with equation \(\mathrm { y } = 2 \mathrm { x } - \mathrm { x } ^ { 2 }\) 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 \(\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x < U _ { n }\), where $$U _ { n } = \left( 1 + \frac { 1 } { n } \right) \left( \frac { 2 } { 3 } - \frac { 1 } { 6 n } \right) .$$
2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x\).
3. Show that \(\lim _ { n \rightarrow \infty } \left( \mathrm { U } _ { n } - \mathrm { L } _ { \mathrm { n } } \right) = 0\).

5 \includegraphics[max width=\textwidth, alt={}, center]{114be67d-a57f-4c36-8f1c-974a2719c1f1-08_663_1152_260_452} The diagram shows the curve with equation \(\mathrm { y } = 2 \mathrm { x } - \mathrm { x } ^ { 2 }\) 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 \(\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x < U _ { n }\), where $$U _ { n } = \left( 1 + \frac { 1 } { n } \right) \left( \frac { 2 } { 3 } - \frac { 1 } { 6 n } \right) .$$
2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x\).
3. Show that \(\lim _ { n \rightarrow \infty } \left( \mathrm { U } _ { n } - \mathrm { L } _ { \mathrm { n } } \right) = 0\).

4 \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-08_408_1433_296_315} The diagram shows the curve with equation \(y = x ^ { - 2 }\) for \(2 \leqslant x \leqslant N\) together with a set of ( \(N - 2\) ) rectangles of unit width.

1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } } > \frac { 3 } { 2 } - \frac { 1 } { N } + \frac { 1 } { N ^ { 2 } }$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-08_2718_35_141_2012}
2. Use a similar method to find, in terms of \(N\), an upper bound for \(\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } }\).
3. Deduce lower and upper bounds for \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }\).

8 \includegraphics[max width=\textwidth, alt={}, center]{5b43cb39-7560-4484-ba6f-17303e986f47-10_369_1531_260_306} The diagram shows the curve \(\mathrm { y } = \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } }\) for \(x \geqslant 0\), together with a set of \(n\) rectangles of unit width. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r ^ { 2 } + r + 1 } } < \ln \left( \frac { 1 } { 3 } + \frac { 2 } { 3 } n + \frac { 2 } { 3 } \sqrt { n ^ { 2 } + n + 1 } \right)$$

6 \includegraphics[max width=\textwidth, alt={}, center]{323ac7a5-4690-441d-87fc-325a393098fa-10_585_1349_258_358} The diagram shows the curve \(\mathrm { y } = \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + 2 \mathrm { x } } }\) for \(x > 0\), together with a set of \(( n - 1 )\) rectangles of unit
width. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r ^ { 2 } + 2 r } } < \ln \left( n + 1 + \sqrt { n ^ { 2 } + 2 n } \right) + \frac { 1 } { 3 } \sqrt { 3 } - \ln ( 2 + \sqrt { 3 } )$$

6 \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-12_533_1532_278_264} The diagram shows the curve with equation \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(N\) rectangles each of width \(\frac { 1 } { N }\).

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } \left( \frac { 1 } { 2 } \right) ^ { x } \mathrm {~d} x > L _ { N }\), where $$L _ { N } = \frac { 1 } { 2 N \left( 2 ^ { \frac { 1 } { N } } - 1 \right) }$$ \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-12_2717_38_109_2009}
2. Use a similar method to find, in terms of \(N\), an upper bound \(U _ { N }\) for \(\int _ { 0 } ^ { 1 } \left( \frac { 1 } { 2 } \right) ^ { x } \mathrm {~d} x\).
3. Find the least value of \(N\) such that \(U _ { N } - L _ { N } \leqslant 10 ^ { - 3 }\).
4. Given that \(\int _ { 0 } ^ { 1 } \left( \frac { 1 } { 2 } \right) ^ { x } \mathrm {~d} x = \frac { 1 } { 2 \ln 2 }\) ,use the value of \(N\) found in part(c)to find upper and lower bounds for \(\ln 2\) .

6 \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-12_533_1532_278_264} The diagram shows the curve with equation \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(N\) rectangles each of width \(\frac { 1 } { N }\).

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } \left( \frac { 1 } { 2 } \right) ^ { x } \mathrm {~d} x > L _ { N }\), where $$L _ { N } = \frac { 1 } { 2 N \left( 2 ^ { \frac { 1 } { N } } - 1 \right) }$$ \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-12_2717_38_109_2009}
2. Use a similar method to find, in terms of \(N\), an upper bound \(U _ { N }\) for \(\int _ { 0 } ^ { 1 } \left( \frac { 1 } { 2 } \right) ^ { x } \mathrm {~d} x\).
3. Find the least value of \(N\) such that \(U _ { N } - L _ { N } \leqslant 10 ^ { - 3 }\).
4. Given that \(\int _ { 0 } ^ { 1 } \left( \frac { 1 } { 2 } \right) ^ { x } \mathrm {~d} x = \frac { 1 } { 2 \ln 2 }\) ,use the value of \(N\) found in part(c)to find upper and lower bounds for \(\ln 2\) .

4 \includegraphics[max width=\textwidth, alt={}, center]{6ff1b572-4cd8-433d-ba16-ffc8cda44476-06_545_958_264_552} The diagram shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } }\) fo \(x > 0\) tg th rwith a set \(6 ( n - 1 )\) rectab es 6 in t witd h

1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } } < \frac { 2 n - 1 } { n }$$
2. Use a similar method to find, in terms of \(n\), a low er \(\mathbf { H }\)
   • \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{268b605f-eb86-40df-946a-210da1355e83-2_686_967_998_589} The diagram shows the curve with equation \(y = \mathrm { e } ^ { x ^ { 2 } }\), for \(0 \leqslant x \leqslant 1\). The region under the curve between these limits is divided into four strips of equal width. The area of this region under the curve is \(A\).

1. By considering the set of rectangles indicated in the diagram, show that an upper bound for \(A\) is 1.71 .
2. By considering an appropriate set of four rectangles, find a lower bound for \(A\).

3 \includegraphics[max width=\textwidth, alt={}, center]{15dd10f9-73d4-4107-bb45-7866f5470572-2_643_787_1621_680} The diagram shows the curve with equation \(y = \sqrt { 1 + x ^ { 3 } }\), for \(2 \leqslant x \leqslant 3\). The region under the curve between these limits has area \(A\).

1. Explain why \(3 < A < \sqrt { 28 }\).
2. The region is divided into 5 strips, each of width 0.2 . By using suitable rectangles, find improved lower and upper bounds between which \(A\) lies. Give your answers correct to 3 significant figures.

6 \includegraphics[max width=\textwidth, alt={}, center]{debf6581-25ff-4692-bdfb-154675a3cdb0-3_608_1134_258_504} The diagram shows the curve \(y = \mathrm { f } ( x )\), defined by $$f ( x ) = \begin{cases} x ^ { x } & \text { for } 0 < x \leqslant 1 , \\ 1 & \text { for } x = 0 . \end{cases}$$

1. By first taking logarithms, show that the curve has a stationary point at \(x = \mathrm { e } ^ { - 1 }\). The area under the curve from \(x = 0.5\) to \(x = 1\) is denoted by \(A\).
2. By considering the set of three rectangles shown in the diagram, show that a lower bound for \(A\) is 0.388 .
3. By considering another set of three rectangles, find an upper bound for \(A\), giving 3 decimal places in your answer. The area under the curve from \(x = 0\) to \(x = 0.5\) is denoted by \(B\).
4. Draw a diagram to show rectangles which could be used to find lower and upper bounds for \(B\), using not more than three rectangles for each bound. (You are not required to find the bounds.)

4 \includegraphics[max width=\textwidth, alt={}, center]{c342b622-a560-46da-9e64-edc4b7b3be93-2_662_1063_986_484} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { x } }\) for \(0 < x \leqslant 1\). A set of ( \(n - 1\) ) rectangles is drawn under the curve as shown.

1. Explain why a lower bound for \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { - \frac { 1 } { x } } \mathrm {~d} x\) can be expressed as $$\frac { 1 } { n } \left( \mathrm { e } ^ { - n } + \mathrm { e } ^ { - \frac { n } { 2 } } + \mathrm { e } ^ { - \frac { n } { 3 } } + \ldots + \mathrm { e } ^ { - \frac { n } { n - 1 } } \right)$$
2. Using a set of \(n\) rectangles, write down a similar expression for an upper bound for \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { - \frac { 1 } { x } } \mathrm {~d} x\).
3. Evaluate these bounds in the case \(n = 4\), giving your answers correct to 3 significant figures.
4. When \(n \geqslant N\), the difference between the upper and lower bounds is less than 0.001 . By expressing this difference in terms of \(n\), find the least possible value of \(N\).

1 \includegraphics[max width=\textwidth, alt={}, center]{cf77e51a-1d3f-423a-be59-96ec60fbeb67-2_568_959_269_593} The diagram shows the curve with equation \(y = \ln ( \cos x )\), for \(0 \leqslant x \leqslant 1.5\). The region bounded by the curve, the \(x\)-axis and the line \(x = 1.5\) has area \(A\). The region is divided into five strips, each of width 0.3 .

1. By considering the set of rectangles indicated in the diagram, find an upper bound for \(A\). Give the answer correct to 3 decimal places.
2. By considering another set of five suitable rectangles, find a lower bound for \(A\). Give the answer correct to 3 decimal places.
3. How could you reduce the difference between the upper and lower bounds for \(A\) ?

6 \includegraphics[max width=\textwidth, alt={}, center]{a80eb21f-c273-4b65-8617-16cdee783305-4_656_1017_251_525} The diagram shows part of the curve \(y = \ln ( \ln ( x ) )\). The region between the curve and the \(x\)-axis for \(3 \leqslant x \leqslant 6\) is shaded.

1. By considering \(n\) rectangles of equal width, show that a lower bound, \(L\), for the area of the shaded region is \(\frac { 3 } { n } \sum _ { r = 0 } ^ { n - 1 } \ln \left( \ln \left( 3 + \frac { 3 r } { n } \right) \right)\).
2. By considering another set of \(n\) rectangles of equal width, find a similar expression for an upper bound, \(U\), for the area of the shaded region.
3. Find the least value of \(n\) for which \(U - L < 0.001\).

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 \(A\).

14 Fig. 14.1 shows the curve with equation \(y = \frac { 1 } { 1 + x ^ { 2 } }\), together with 5 rectangles of equal width. \begin{figure}[h] \end{figure} Fig. 14.2 shows the coordinates of the points \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F . \section*{Fig. 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 \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } \mathrm { dx }\) is 0.7337, correct to \(\mathbf { 4 }\) decimal places.
   [0pt] [2]
2. Use the 5 rectangles shown in Fig. 14.1 and the information in Fig. 14.2 to calculate an upper bound for \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } \mathrm {~d} x\) correct to \(\mathbf { 4 }\) decimal places.
   [0pt] [2]
3. Hence find the length of the interval in which your answers to parts (a) and (b) indicate the value of \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } \mathrm {~d} x\) lies.
   [0pt] [1] Amit uses \(n\) rectangles, each of width \(\frac { 1 } { n }\), to calculate upper and lower bounds for \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } \mathrm {~d} x\), using different values of \(n\). His results are shown in Fig. 14.3.

   \(n\)	10	20	40
   upper bound	0.80998	0.79779	0.79162
   lower bound	0.75998	0.77279	0.77912

   \section*{Fig. 14.3}
4. Find the length of the smallest interval in which Amit now knows \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } \mathrm { dx }\) lies.
5. Without doing any calculation, explain how Amit could find a smaller interval which contains the value of \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } d x\).

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 \(p\).