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QUESTION / ANSWER BOOKLET ENGSCI 211
THE UNIVERSITY OF AUCKLAND
SEMESTER ONE 2020
Campus: City
ENGINEERING SCIENCE
Mathematical Modelling 2
Term Test 2
(Time allowed: TWENTY FOUR hours)
NOTE: This test should take approximately one hour to complete.
This test is made up of TWO sections:
* Section A – Multivariable Calculus and Linear Algebra: Answers are to
be written on paper, and a PDF, DOC or DOCX file with your scanned
or photographed answers is to be submitted to the Canvas Assignment
dropbox for Section A.
* Section B – Data Analysis: Answers to each part (including R commands
and output) are to be typed into the R Notebook or Word template for
this section. Submit your knitted Notebook as a PDF, DOCX or HTML
file or your final Word document with your answers to Section B into the
Canvas Assignment dropbox for Section B.
Show all working. Marks may be deducted if key steps are missing.
An appendix containing R output for Section B, and the formula sheet, is
attached to the back of this booklet.
Surname: Forename(s):
University of Auckland ID number:
University of Auckland Username:
Question 1 2 3 4 Total
Out o

f 8 7 10 10 35
Page 1 of 8
QUESTION / ANSWER BOOKLET
UoA Username:
ENGSCI 211
SECTION A – Multivariable Calculus and Linear Algebra
Question 1 – Multivariable Integration (8 marks)
Consider the following double integral:∫ 16
4
∫ √y
1
f(x, y) dx dy
(a) Sketch the region of integration. Label all intercepts and axes. (2 marks)
1 2 3 4
4
8
12
16
x =
√
y y = x2
x
y
(b) Re-write the integral after changing the order of integration. (3 marks)
Between x = 1 to x = 2 (rectangular bit):
∫ 2
1
∫ 16
4
f(x, y) dy dx
Between x = 2 to x = 4 (curved bit):
∫ 4
2
∫ 16
x2
f(x, y) dy dx
(c) We wish to integrate the function g(x, y) = x2 + y2 + 2 in the area bounded by a circle
of radius 5 centered at the origin, and where y ≥ 0.
Re-write this integral in polar coordinates.
Do NOT evaluate this integral – there are no marks for this. (3 marks)
Substituting x = r cos θ and y = r sin θ gives:∫∫
(x2 + y2 + 2) dx dy =
∫ pi
0
∫ 5
0
(r2 + 2)r dr dθ
Page 2 of 8
QUESTION / ANSWER BOOKLET
UoA Username:
ENGSCI 211
Question 2 – Linear Algebra (7 marks)
(a) Find the PLU factorisation of:
A =
2 3 02 3 2
4 7 3