Question

The cam lobe from a car’s 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)=1.5-1.5x⁴,

where y is the vertical height in centimetres above the x-axis. The x-axis runs from x=-1cm to x=1 cm, with the origin at its centre. The lower half of the profile is a semicircle of radius 1 cm.

    Figure 3 Cross-section of the camshaft in question 3

 

a.     By splitting the area of the cam above the x-axis into a series of 20 rectangular strips, as shown below in Figure 4, calculate an approximate value for the area of the cam above the x-axis. 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. 

                      Figure 4 Division of upper camshaft to 20 strips

 

b.     Carry out the same calculation of the area of the cam lobe (above the x-axis), this time using integration to calculate the area of the cam lobe above the x-axis. 

Show all your steps for integration.

c.     Calculate the total cross-section area of the cam including its lower half, using the value you obtained in part (b).