The cam lobe in an engine, shown below in Figure 3, is cut from a steel...

 

 

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The cam lobe in an engine, shown below in Figure 3, is cut from a steel plate. The upper half of its profile is described by the function y(x) = 2.5-2.5x4 where y is the vertical height in centimetres above the x-axis. The x-axis runs from x = -1 cm to x = 1 cm, with the origin at its centre. The lower half of the profile is given by y (x) = -1.5 +1.5x². 2.5 1.5 1 0.5 0 -0.5 -1 -1.5 -1 -0.5 0 Figure 3 Cross-section of the cam shaft in question 3 0.5 0 1 0.5 1.5 2 1 2.5 1 3 Figure 5 Displacement of an oscillating spring in a vehicle. 3.5 4 a. By splitting the area of the cam into a series of 20 rectangular strips, as shown in Figure 4, calculate an approximate value for area of the cam. The function y(x) is even, that is to say y(-x) = y(x); this can save you a little work. Work throughout to two decimal places and quote your estimate to one decimal place. (18 marks) Hint: To determine the overall length of each strip, you will need to calculate the length that lies above the x-axis and the length that lies below the x-axis and add them together. Estimating the length from Figure 4 will not be accurate enough. 2.5 2 1.5 0.5 0 -0.5 -1 -1.5- -0.5 0 0.5 1 b. Carry out the same calculation of the area of the cam lobe, this time using integration to calculate the area of the cam. Show all your steps for integration. (8 marks) The cam lobe in an engine, shown below in Figure 3, is cut from a steel plate. The upper half of its profile is described by the function y(x) = 2.5-2.5x4 where y is the vertical height in centimetres above the x-axis. The x-axis runs from x = -1 cm to x = 1 cm, with the origin at its centre. The lower half of the profile is given by y (x) = -1.5 +1.5x². 2.5 1.5 1 0.5 0 -0.5 -1 -1.5 -1 -0.5 0 Figure 3 Cross-section of the cam shaft in question 3 0.5 0 1 0.5 1.5 2 1 2.5 1 3 Figure 5 Displacement of an oscillating spring in a vehicle. 3.5 4 a. By splitting the area of the cam into a series of 20 rectangular strips, as shown in Figure 4, calculate an approximate value for area of the cam. The function y(x) is even, that is to say y(-x) = y(x); this can save you a little work. Work throughout to two decimal places and quote your estimate to one decimal place. (18 marks) Hint: To determine the overall length of each strip, you will need to calculate the length that lies above the x-axis and the length that lies below the x-axis and add them together. Estimating the length from Figure 4 will not be accurate enough. 2.5 2 1.5 0.5 0 -0.5 -1 -1.5- -0.5 0 0.5 1 b. Carry out the same calculation of the area of the cam lobe, this time using integration to calculate the area of the cam. Show all your steps for integration. (8 marks)