Consider the following functions below given by: I x cos x dx 0, n =...

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Consider the following functions below given by: I x cos x dx π 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 ≤x≤3.5. (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 0≤x≤3 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,-1≤x≤ 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) Consider the following functions below given by: I x cos x dx π 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 ≤x≤3.5. (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 0≤x≤3 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,-1≤x≤ 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)

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111 i Leftend Rule Using the Leftend Rule the area of the given function is calculated as follows A ... View the full answer