Module MA1132 (Frolov), Advanced Calculus
Tutorial Sheet 8
To be solved during the tutorial session Thursday/Friday, 17/18 March 2016
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 in the first quadrant bounded by the curves r = 2a sin θ, and
θ = φ ≤ π/2, where r and θ are the polar coordinates: x = r cos θ, y = r sin θ.
(a) What is the curve r = 2a sin θ?
(b) What is the curve θ = φ?
(c) Sketch the region R.
(d) Find the area of R.
2. Find the volume V of the solid bounded by the planes x = 0, y = 0, z = 0, the cylinder
x2 + y 2 = a2 and the hyperbolic paraboloid z = xy in the first octant.
3. Find a parametric representation (different from the generalised stereographic projection)
of
(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 = −1 .
a2 a2
c
4. Consider the portion of the surface (y − 2)2 + (z + 1)2 = 12 that is above the rectangle
√
√
3
R = {(x, y) : −
≤ x ≤ 3 , −1 ≤ y ≤ 5}.
2
(a) What is the surface?
(b) Sketch the projection of the portion onto the xy-plane.
(c) Use double integration to find the area of the portion.
Show the details of your work.
5. Find the area of the portion of the hyperbolic paraboloid z = xy that is in the first octant
x, y, z ≥ 0 and inside the cylinder x2 + y 2 = a2 .
1