Mathematical Methods for Engineers 2 (MATH 1064)
SP1 2022 Module 2 Assignment 2
Due at 11.59pm on Monday 21 February
Total: 35 marks
Late submissions without approval of the Course coordinator attract a penalty of 10% per day, up
to 40%. After 4 days the work won’t be accepted.
You are required to submit your handwritten or typed solutions as a single PDF
file using the online link on the course webpage. If you submit handwritten solutions,
make sure to write legibly and make a good scan. If you use MS Word, type all mathematical
expressions using Equation editor. Both handwritten and typed mathematical expressions must
be notationally correct and clearly presented. Marks will be deducted for poorly presented
solutions. Show your working out.
1. Solve the following differential equations
(a) y′ tanx+ y = sinx cosx
(b) y′′ + y′ − 2y = e−2t, y(0) = 0, y′(0) = 2
(c) y′′ + 4y = sinx
(6+9+6=21 mark)
2. A flat region R is bounded by the curve y = 4 − x2 and by x- and y-axes in the first
quadrant.
(a) Make an accurate sketch of the flat region R.
(b) When the region R is revolved about the y-axis it forms a solid object. Assume that
the solid has a uniform density ρ.
Find the exact volume V of the solid by
i. Slicing it horizontally (stacked disks method). Start with finding dV .
ii. By using cylindrical shells method. Start with dV .
(c) If the point C(x¯, y¯) is the centroid of the solid, explain why x¯ = 0.
(d) Use the results of the previous sections to find the exact height of the centroid, y¯.
(e) Calculate the moment of inertia of the solid about y-axis, Iy.
Show that Iy = MR
2
gyr, where M is the mass of the solid and Rgyr is the radius of
gyration.
(1+(3+3)+1+3+3=14 marks)