Use the trapezium rule with 5 ordinates to find an estimate for the area of the shaded region - AQA - A-Level Maths Pure - Question 14 - 2022 - Paper 1

Using the trapezium rule with 5 ordinates means dividing the interval ([1, 4]) into 4 equal parts. The width of each part ( h ) is calculated as: [ h = \frac{b - a}{n} = \frac{4 - 1}{4} = 0.75 ]

Now, we compute the y-values (ordinates) at each x value:

• ( y_1 = (2(1) - 8) \ln(1) = 0 )
• ( y_2 = (2(1.75) - 8) \ln(1.75) \approx -2.51827 )
• ( y_3 = (2(2.5) - 8) \ln(2.5) \approx -2.74887 )
• ( y_4 = (2(3.25) - 8) \ln(3.25) \approx -1.76789 )
• ( y_5 = (2(4) - 8) \ln(4) = 0 )

Using the trapezium rule formula: [ A = \frac{h}{2} \left( y_1 + 2y_2 + 2y_3 + 2y_4 + y_5 \right) ] Substituting the values: [ A = \frac{0.75}{2} \left( 0 + 2(-2.51827) + 2(-2.74887) + 2(-1.76789) + 0 \right) ] Calculating: [ A = \frac{0.75}{2} \left( -5.03654 - 5.49774 - 3.53578 \right) \approx - 5.28 ]

Thus, the estimated area of the shaded region, correct to three significant figures, is ( 5.28 ).