Consider the following functions below given by: I x cos x dx 0, n = 4. 1.1.1 Calculate the area using (1) the Left-end rule (3) (ii) the Midpoint Rule and (3) (iii) the right-end point rule (3) 1.1.2 Calculate the error and error percentage approximation for all the three functions in 1.1 (3). 1.2 The equation of the curve is given by y = 8x - 2x 2, 0.5 x 3.5. with n= 6. Calculate 1.2.1 the length of the curve using the Midpoint Rule. (4) 1.2.2 the area of the curve under the defined region 0.5 x3.5. (4) 1.3 Find the volume of the solid formed by revolving the region bounded by the graphs of y = x 2 + 1, y = 0, x = 0 and x = 1 about the y-axis. (5) QUESTION 2 [18] 2.1 A curve C has equation y = x 1 2 -1 3 x 23, x 0. Show that the area of the surface generated when the arc of C for which 0x3 is rotated through 277 radians about the x-axis is 37 square units. (4) 2.2 Find the area of the surface formed when f(x) = x 2 between 0 and 2 is rotated around the y-axis. 2.3 The curve y=4-x2,-1x 1, is an are of the circle x 2 + y 2 = 4. Find the area of the surface obtained by rotating this arc about the 2.3.1 x- axis, (4) 2.3.2 y-axis (4) 2.4 Find the volume of the solid formed by revolving the region bounded by the graph of y = sin x and the x-axis (0 x 7) about the x-axis. (6) QUESTION 3 [17] 3.1 Find the volume of the solid obtained by rotating the region bounded by y=x 3, y = 8, and x = 0 about the y-axis. (6) 3.2 Find the volume of the solid formed by revolving the region bounded by f(x) = 2 - x 2 and g(x) = 1 about the line y = 1. (5) 3.3 A manufacturer drills a hole through the centre of a metal sphere of radius 5 m. The hole has a radius of 3 m. What is the volume of the resulting metal ring? (6)