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The continuous curve \newlineCC has equation \newliney=f(x)y=f(x). \newlineA table of values of \newlinexx and \newlineyy for \newliney=f(x)y=f(x) is shown below.\newlinexx\newline4.04.0\newline4.24.2\newline4.44.4\newline4.64.6\newliney=f(x)y=f(x)00\newliney=f(x)y=f(x)11\newlineyy\newliney=f(x)y=f(x)33\newliney=f(x)y=f(x)44\newliney=f(x)y=f(x)55\newliney=f(x)y=f(x)66\newliney=f(x)y=f(x)77\newliney=f(x)y=f(x)88\newlineUse the trapezium rule with all the values of \newlineyy in the table to find an approximation for\newlinexx00\newlinegiving your answer to xx11 decimal places.

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Q. The continuous curve \newlineCC has equation \newliney=f(x)y=f(x). \newlineA table of values of \newlinexx and \newlineyy for \newliney=f(x)y=f(x) is shown below.\newlinexx\newline4.04.0\newline4.24.2\newline4.44.4\newline4.64.6\newliney=f(x)y=f(x)00\newliney=f(x)y=f(x)11\newlineyy\newliney=f(x)y=f(x)33\newliney=f(x)y=f(x)44\newliney=f(x)y=f(x)55\newliney=f(x)y=f(x)66\newliney=f(x)y=f(x)77\newliney=f(x)y=f(x)88\newlineUse the trapezium rule with all the values of \newlineyy in the table to find an approximation for\newlinexx00\newlinegiving your answer to xx11 decimal places.
  1. Identify width of trapezium: Identify the width of each trapezium hh. Since the xx-values increase by 0.20.2, h=0.2h = 0.2.
  2. Apply trapezium rule: Apply the trapezium rule: f(x)dx(h2)×(y0+2y1+2y2+2y3+2y4+y5)\int f(x)dx \approx (\frac{h}{2}) \times (y_0 + 2y_1 + 2y_2 + 2y_3 + 2y_4 + y_5), where y0y_0 to y5y_5 are the y-values from the table.
  3. Substitute values into formula: Substitute the values into the trapezium rule formula: 0.22\frac{0.2}{2} * 9.2+2×8.4556+2×3.8512+2×5.0342+2×7.8297+8.69.2 + 2\times8.4556 + 2\times3.8512 + 2\times5.0342 + 2\times7.8297 + 8.6.
  4. Calculate sum inside parentheses: Calculate the sum inside the parentheses: 9.2+2×8.4556+2×3.8512+2×5.0342+2×7.8297+8.6=9.2+16.9112+7.7024+10.0684+15.6594+8.69.2 + 2 \times 8.4556 + 2 \times 3.8512 + 2 \times 5.0342 + 2 \times 7.8297 + 8.6 = 9.2 + 16.9112 + 7.7024 + 10.0684 + 15.6594 + 8.6.
  5. Add up values: Add up the values: 9.2+16.9112+7.7024+10.0684+15.6594+8.6=68.14149.2 + 16.9112 + 7.7024 + 10.0684 + 15.6594 + 8.6 = 68.1414.
  6. Multiply sum by factor: Multiply the sum by (h/2)(h/2): (0.2/2)×68.1414=0.1×68.1414(0.2/2) \times 68.1414 = 0.1 \times 68.1414.
  7. Calculate final result: Calculate the final result: 0.1×68.1414=6.814140.1 \times 68.1414 = 6.81414.

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