School of Mathematics and Statistics
MAST20009 Vector Calculus, Semester 1 2021
Assignment 3 and Cover Sheet
Student Name Student Number
Tutor’s Name Tutorial Day/Time
Submit your assignment via Gradescope accessed via the MAST20009 Canvas page
before 12pm on Tuesday 4 May.
• This assignment is worth 5% of your final MAST20009 mark.
• Assignments must be neatly handwritten in blue or black pen on A4 paper or may be typeset using
LATEX. Diagrams can be drawn in pencil.
• You must complete the plagiarism declaration on the LMS before submitting your assignment.
• Full working must be shown in your solutions.
• Marks will be deducted for incomplete working, insufficient justification of steps, incorrect
mathematical notation and for messy presentation of solutions.
1. Let the path C traverse half of the ellipse 4x2 + 9y2 = 36 from (3, 0) to (−3, 0) in a clockwise
direction.
(a) Write down a parametrisation for C in terms of an increasing parameter t.
(b) Using part (a), determine the work done by the force
F(x, y) = 2yi+ 5xj
to move a particle along C.
2. Let a and h be positive real numbers. Let Σ be the surface in R3 given by
4ax = y2 + z2.
(a) Find the volume of the region enclosed between Σ and the plane x = h.
(b) Show that the surface area of the part of Σ with x ≤ h is
8pi
√
a
3
(
(a+ h)3/2 − a3/2)) .
3. Let n be a positive integer. Let f : [0, 1]→ R be a continuous function. Show that∫ 1
0
∫ 1
0
f(xy)xn+1y(1− y)n−1 dx dy = 1
n
∫ 1
0
t(1− t)nf(t) dt.
[Hint: Perform the change of variables s = x(1− y), t = xy].
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