Module MA1132 (Frolov), Advanced Calculus
Homework Sheet 7
Each set of homework questions is worth 100 marks
Due: at the beginning of the tutorial session Thursday/Friday, 17/18 March 2016
Name:
You may use Mathematica to sketch the integration regions and solids, and to check the
results of integration.
1. Evaluate the double integral
ZZ
1
dxdy ,
x+y
I=
R
where R is the region enclosed by the lines x = 2, y = x, and the hyperbola xy = 1.
2. Sketch the integration region R and reverse the order of integration
(a)
√
Z
7/2
7+12y−4y 2
2+
Z
√
−1/2
2−
(b)
1
Z
f (x, y)dxdy
√
Z
3−x2
f (x, y)dydx
0
(1)
7+12y−4y 2
(2)
x2 /2
3. Sketch the integration region R and write the expression as one repeated integral by
reversing the order of integration
Z 6Z 3
Z 8 Z 9−x
f (x, y)dydx +
f (x, y)dydx
(3)
2
9/(x+1)
4. Prove the Dirichlet formula
Z bZ
6
x
9/(x+1)
Z bZ
f (x, y)dydx =
a
and use it to prove that
Z xZ
a
(t1 − t)
a
a
f (x, y)dxdy ,
a
t1
n−1
b
(4)
y
1
f (t)dtdt1 =
n
Z
x
(x − t)n f (t)dt .
(5)
a
5. Find the volume V of the solid bounded by
(a) the planes x = 0, y = 0, z = 0, 2x + y = 1, and the surface z = x + y 2 + 1.
(b) the planes x = 1, z = 0, the parabolic cylinder x − y 2 = 0, and the paraboloid
z = x2 + y 2 .
1
(c) 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.
(d) the planes z = zmin , z = λx + µy + h (λ > 0, µ > 0, h > 0), and the elliptic cylinder
2
x2
+ yb2 = 1, where zmin is the z-coordinate of the lowest point on the intersection of
a2
2
2
z = λx + µy + h (λ > 0, µ > 0, h > 0) and xa2 + yb2 = 1.
Bonus questions (each bonus question is worth extra 25 marks)
1. The Viviani solid is the solid inside the sphere x2 + y 2 + z 2 = a2 and inside the cylinder
x2 + y 2 = ax. Sketch the Viviani solid, and find its volume.
2. Evaluate the integrals
ZZ
ZZ
ydxdy ,
I1 =
R
I2 =
xdxdy ,
(6)
R
where R is the region enclosed by an arc of the cycloid
x = a(t − sin t) ,
y = a(1 − cos t) ,
and the x-axis.
2
0 ≤ t ≤ 2π ,
(7)