Module MA1132 (Frolov), Advanced Calculus
Homework Sheet 8
Each set of homework questions is worth 100 marks
Due: at the beginning of the tutorial session Thursday/Friday, 24/25 March 2016
Name:
You may use Mathematica to sketch the integration regions and solids, and to check the
results of integration. Use polar coordinates to evaluate double integrals.
1. Consider the region R that is inside the curve r = 4 cos θ and outside the curve (lemniscate) r2 = −8 cos 2θ , where r and θ are the polar coordinates: x = r cos θ, y = r sin θ.
(a) What is the curve r = 4 cos θ?
(b) Sketch the region R.
(c) Find the area of R.
2. Find the volume V of the solid bounded by the planes x = 0, y = 0, z = 0, the cylinders
az = x2 , a > 0, x2 + y 2 = b2 , and located in the first octant x ≥ 0, y ≥ 0, z ≥ 0.
3. Consider the solid inside the surface r2 + z 2 = 4 and outside the surface r = 2 sin θ.
(a) What is the surface r2 + z 2 = 4?
(b) What is the surface r = 2 sin θ?
(c) Sketch the solid. Sketch the intersection of the solid with the xy-plane
(d) Find the volume V of the solid
4. Find a parametric representation (different from a generalised stereographic projection)
of
(a) the elliptic cone
r
z=
x2 y 2
+ 2.
a2
b
(b) the hyperboloid of one sheet
x2 y 2 z 2
+ 2 − 2 = 1.
a2
b
c
5. Read about the stereographic projection in Wikipedia and generalise it to parametrise
(a) the ellipsoid
x2 y 2 z 2
+ 2 + 2 = 1.
a2
b
c
(b) the hyperboloid of two sheets
x2 y 2 z 2
+ 2 − 2 = −1 .
a2
b
c
1
(c) the hyperboloid of one sheet
−
x2 y 2 z 2
+ 2 + 2 = 1.
a2
b
c
6. Find the area of the portion of the elliptic paraboloid z = c −
2
2
cylinder xa2 + yb2 = c2 .
x2
2a
−
y2
2b
that is inside the
7. Find the area of the portion of the sphere x2 + y 2 + z 2 = a2 that is inside the cylinder
x 2 + y 2 = b2 .
8. Find the area of the portion of the sphere x2 + y 2 + z 2 = a2 that is outside the cylinder
x2 + y 2 = ay.
9. Find the area of the portion of the cylinder x2 + y 2 = ay that is inside the sphere
x 2 + y 2 + z 2 = a2 .
Bonus questions (each bonus question is worth extra 25 marks)
1. Find the area of the portion of the sphere x2 + y 2 + z 2 = 2az that is inside the cone
z 2 = Ax2 + By 2 .
2. Prove that the area of the surface
(x2 + y 2 + z 2 )2 = α2 x2 + β 2 y 2 + γ 2 z 2
coincides with the area of the ellipsoid
x2 y 2 z 2
+ 2 + 2 = 1,
a2
b
c
if one takes
a=
βγ
,
α
b=
2
γα
,
β
c=
αβ
.
γ
(1)