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MATH2001/MATH7000 ASSIGNMENT 3 SEMESTER 2 2023
Due at 5:00pm 6 October. Marks for each question are shown. Total marks: 25
Submit your assignment online via the Assignment 3 Gradescope submission link in
Blackboard.
(1) (4 marks) Find the volume of the solid enclosed by the surface
z = x− xy2 + 12
and the planes z = 0, x = 0, x = 3 and y = ±2.
(2) (5 marks) Evaluate the double integral∫∫
D
y dA,
where D is the region bounded by y = x− 2, x = y2.
(3) (6 marks) Use polar coordinates to evaluate the following integrals:
(a)
∫ a
0
∫ √a2−y2
−
√
a2−y2
(2x + y) dx dy.
(b)
∫ 0
−1
∫ √4−y2
−√3y
xy2 dx dy.
(4) (5 marks) The average value of a function f(x, y, z) over a solid region E is
defined as
fave =
1
V (E)
∫∫∫
E
f(x, y, z) dV
where V (E) is the volume of E. Find the average value of the function
f(x, y, z) = xyz over the cube with side length L that lies in the first octant
with one vertex at the origin and edges parallel to the coordinate axes.
(5) (5 marks) Find the mass and centre of mass of a solid hemisphere of radius R if
the density at any point is proportional to its distance from the base.