Algebra Question

The a-s-s-e-s-s-m-e-n-t will be 5 calculation questions. Please see the questions shown in the screenshot. I will send you all info after being hired, eg PPTs, student access etc. Please send a draft in 12hrs -1 day time, day 2, and day 3 as well. + Will need to draft some questions to ask the teacher and revise base on feedback (Send bk ard in 1 day max)

MATH2001/MATH7000 ASSIGNMENT 3
SEMESTER 2 2022
Due at 4:00pm 7 October. Marks for each question are shown. Total marks: 20
Submit your assignment online via the assignment 3 submission link in Blackboard.
(1) (2 marks) Find the volume of the solid enclosed by the surface
z = 1 + x2 yey
and the planes z = 0, x = ±1, y = 0, and y = 1.
(2) (4 marks) Evaluate the double integral
ZZ
y 2 exy dA,
D
where D is the region bounded by y = x, y = 4, x = 0.
(3) (4 marks) Use polar coordinates to evaluate the following integrals:
Z
Z √
1−y 2
1/2
(a)
√
0
xy 2 dx dy.
3y
Z 2 Z √2x−x2 p
(b)
x2 + y 2 dy dx.
0
0
(4) (5 marks) The average value of a function f (x, y, z) over a solid region E is
defined as
ZZZ
1
f (x, y, z) dV
fave =
V (E)
E
where V (E) is the volume of E. Find the average height of the points in the solid
hemisphere x2 + y 2 + z 2 ≤ 1, z ≥ 0.
(5) (5 marks) Let H be a solid hemisphere of radius a with constant density.
(a) Find the centroid of H.
(b) Find the moment of inertia of H about a diameter of its base.