Option 1: For students enrolled in Civil, Construction, Mechanical, Coastal, or Environmental Engineering.

A circular shaft has a diameter, d (m), that varies with axial position, x (m), along the shaft (from the support position at x = 0 m). The variation is given by d = 0.02e^-x (1+x² ) where 0 ≤ x ≤ 3 m. An axial load P = 40kN is applied to the end of the shaft, whose Modulus Elasticity E = 2*10¹¹ Nm^-2.

The axial elongation of the ?L (m) of the shaft in the range of values of 0 ≤ x ≤ 3 m is given by: ?L = P/E ∫4dx/(πd²).

a) Using MATALB, plot the shaft diameter with axial position. Use appropriate labels.

b) Use MATLAB integration techniques to determine the total Axial Elongation.

c) Explain the theory of Simpson 1/3 rule for integration in detail. Use appropriate examples to demonstrate how it works and how the integration result is computed.

Option 2: For students enrolled in Electrical, Electronic, or Mechatronic Engineering.

The effective average DC equivalent value of an alternating current is given by the formula:

IRMS = √(T^-1∫[i(t)]²dt)

where i(t) is the instantaneous current (A) and T is the waveform period (s) and the lower and the upper limits of the integration are 0 and T. The current within a circuit (which replicates every 5 ms) is given by:

i(t) = 10e^-t/5sin(2πt/5) (mA)

a) Using MATALB, plot the circuit current i(t) over four periods. Use appropriate labels.

b) Use MATLAB integration techniques to determine the equivalent DC current (RMS) within the circuit.

c) Explain the theory of Simpson 1/3 rule for integration in detail. Use appropriate examples to demonstrate how it works and how the integration result is computed.