Module MA1132 (Frolov), Advanced Calculus
Tutorial Sheet 7
To be solved during the tutorial session Thursday/Friday, 10/11 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 p
4x2 − y 2 dxdy ,
I=
R
where R is the region enclosed by the lines y = 0, y = x, and x = 1.
2. Sketch the integration region R and reverse the order of integration
(a)
4
Z
12x
Z
f (x, y)dydx
(1)
f (x, y)dydx
(2)
3x2
0
(b)
Z
0
1
Z
3x
2x
3. Sketch the integration region R and write the expression as one repeated integral by
reversing the order of integration
Z 1Z y
Z 3Z 1
f (x, y)dxdy +
f (x, y)dxdy
(3)
0
y 2 /9
1
y 2 /9
4. Find the volume V of the solid bounded by
(a) the planes x = 0, y = 0, z = 0, the cylinder x2 + y 2 = a2 and the hyperbolic
paraboloid z = xy in the first octant.
(b) the planes y = 0, z = 0, y = ab x, and the elliptic cylinder
1
x2
a2
+
z2
c2
= 1.