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.