MAT2615
ASSIGNMENT 4 2025
UNIQUE NO.
DUE DATE: 2025
, lOMoARcPSD|21997160
1. (Section 14.6, Chapter 17)
p
Let V be a region
p in R 3
bounded above by the hemisphere z = 1 − x 2 − y 2 and below by
the cone z = x 2 + y 2 − 1. Let S be the surface of V (consisting of the hemisphere on top
and the paraboloid below).
(a) Compute the volume of V using spherical coordinates. (10)
(b) Sketch S and the XY-projection of S. (3)
(c) Use a surface integral to evaluate the area of S. (8)
[21]
2. (Chapter 17,Section 19.1)
Consider the intersection R between the two circles x2 + y 2 = 2 and (x − 2) 2 + y 2 = 2.
y
R x
(a) Find a 2-dimensional vector field F = (M(x , y ), N(x , y )) such that
∂N ∂M
− = 1.
∂x ∂y
(3)
ZZ
(b) Using this F and Green’s theorem (Theorem 19.1.1), write the area integral 1 dA
R
as a line integral.
[Hint: Any function y = f (x ) can be parametrised by r (t) = t , f (t ). This can be used to
parametrise the lines]. (4)
(c) Using this line integral, find the area of R. (9)
[16]
3. (Sections 16.4,19.2)
Consider the surface
p
S = (x , y , z): z = 4 − x 2 − y 2; z ≥ 0
oriented upward. Evaluate the flux integral
ZZ
(curl F ) · n dS
S
2
ASSIGNMENT 4 2025
UNIQUE NO.
DUE DATE: 2025
, lOMoARcPSD|21997160
1. (Section 14.6, Chapter 17)
p
Let V be a region
p in R 3
bounded above by the hemisphere z = 1 − x 2 − y 2 and below by
the cone z = x 2 + y 2 − 1. Let S be the surface of V (consisting of the hemisphere on top
and the paraboloid below).
(a) Compute the volume of V using spherical coordinates. (10)
(b) Sketch S and the XY-projection of S. (3)
(c) Use a surface integral to evaluate the area of S. (8)
[21]
2. (Chapter 17,Section 19.1)
Consider the intersection R between the two circles x2 + y 2 = 2 and (x − 2) 2 + y 2 = 2.
y
R x
(a) Find a 2-dimensional vector field F = (M(x , y ), N(x , y )) such that
∂N ∂M
− = 1.
∂x ∂y
(3)
ZZ
(b) Using this F and Green’s theorem (Theorem 19.1.1), write the area integral 1 dA
R
as a line integral.
[Hint: Any function y = f (x ) can be parametrised by r (t) = t , f (t ). This can be used to
parametrise the lines]. (4)
(c) Using this line integral, find the area of R. (9)
[16]
3. (Sections 16.4,19.2)
Consider the surface
p
S = (x , y , z): z = 4 − x 2 − y 2; z ≥ 0
oriented upward. Evaluate the flux integral
ZZ
(curl F ) · n dS
S
2
