The table below shows corresponding values of x and y for y= X y Figure 3 2 sin 2x Figure 3 shows a sketch of the curve with equation y = (1+cos x)' The finite region R, shown shaded in Figure 3, is bounded by the curve and the x-axis. 0 0 8 0-0137 S 2 sin 2x R (1 + cos x) where k is a constant. dx T 4 = 1.17157 3 RIN 1.02280 45-uzu (U) du u TC 2 sin 2x (1 + cos x) (a) Complete the table above giving the missing value of y to 5 decimal places. 4 ln(1 + cos x) - 4cosx + k X (1) (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 4 decimal places. (3) Cost=41 H(47) H(47) = sinx = -zu (c) Using the substitution u = 1 + cos x, or otherwise, show, that 2sin 2x=4sinx00SX du dx = 1-sinx dx-dy 0x TC 2 BIN 2 0 I-sinx =F=1=24 (5) (d) Hence calculate the error of the estimate in part (b), giving your answer to 2 significant figures. Leave blank (3) 43