C
School of Science (Mathematical Sciences)
ENGINEERING MATHEMATICS
FINAL ASSIGNMENT - QUESTION 1
1. (a) Without using a calculator, determine the following integral:
x2 − 6x + 25
dx.
0 x2 + 6x + 25
(Hint: First write the integrand I(x) as
Z 3
x2 − 6x + 25
ax + b
=1+ 2
2
x + 6x + 25
x + 6x + 25
where a and b are to be determined.)
I(x) =
(b) A steel storage tank for propane gas is to be constructed in the shape of a right
circular cylinder with a hemisphere at each end. Suppose the cylinder has length
` metres and radius r metres.
r
`
(i) Write down an expression for the volume V of the storage tank (in terms of
` and r).
(ii) Write down an expression for the surface area A of the storage tank (in terms
of ` and r).
(iii) Using the result of part (ii), write V as a function of r and A. (That is,
eliminate `.)
(iv) A client has ordered a tank, but can only afford a tank with a surface area
of A = 60 square metres. Given this constraint, write V = V (r).
(v) The client requires the tank to have volume V = 20 cubic metres. Use
Newton’s method, with an initial guess of r0 = 3 to find an approximation
(accurate to three decimal places) to value of r which produces a volume of
20 cubic metres.
(Newton’s method for solving f (r) = 0:
rn+1 = rn −
f (rn )
f 0 (rn )
for n = 0, 1, 2, . . .)
(8 + (1 + 1 + 3 + 1 + 6) = 20 marks)