MATHEMATICAL METHODS FOR ENGINEERS 2
Problem Solving Exercise 2
1. The plane region R is bounded above and on the right by the graph of y = 16 − x2, on
the left by the y-axis, and below by the x-axis. A paraboloid is generated by revolving R
around the y-axis. Then a vertical hole of radius 1 and centred along the y-axis is bored
through the paraboloid. Assume that the solid that remains has a constant density, ρ.
(a) Show that the volume of the body is
225π
2
, by taking horizontal slices.
(b) Find the same volume by the method of infinitessimal annular cylinders.
(c) The centre of mass of this solid lies on the y-axis. Find the y-coordinate of the centre
of mass, y.
(d) Find the moment of inertia of this body rotating about the y-axis.
[ XXXXXXXXXX = 12 marks]
2. (a) Consider the following second order ODE:
d2y
dx2
− 10dy
dx
+ 25y = 0.
Find the general solution by trying solutions of the form y = eλx and using the
characteristic equation.
(b) Using your result from part (a), find the general solution to
d2y
dx2
− 10dy
dx
+ 25y = e5x.
Find the particular solution for which y(0) = 4 and y′(0) = 6.
[3 + 5 = 8 marks]
3. (a) Consider the matrix
A =
1 −2 33 4 −11
1 0 −1
i. Find the reduced row echelon form (RREF) of A.
ii. What is the determinant of A?
iii. Is A invertible?
(b) Consider the following matrix:
A =
1 −2 33k 4 −11
k 0 −1
i. For what value(s) of k does the inverse A−1 not exist?
ii. Find A−1 for the case when k = −1.
[4 + 5 = 9 marks]
4. Consider the following system of 3 equations:
x + 2z = 2
2x + y + 2z = 4
−x − y + z = −1
(a) Write a matrix A and the co
esponding column vector b for which the above system
is Ax = b.
(b) Use Matlab to perform elementary row operations on the augmented matrix [A b ].
From the reduced row echelon form, find the solution(s) to the above system of
equations (if any).
[2 + 4 = 6 marks]
[Total 35 marks]