2S1: Assignment 3
27th January 2009
Due date: Tuesday 10th February
1. Find the average value of the function f (x, y) = x2 y on the rectangular
region R = [1, 3] × [−4, 4] in R2 .
2. Evaluate the following integral by making a sketch of the region of
integration and then reversing the order of integration:
Z 1Z 1
x
e y dy dx
0
x
3. A thin sheet of metal encloses a region D in the first quadrant of the
xy-plane which is bounded by the circles x2 + y 2 = 1 and x2 + y 2 = 9.
The density of the metal sheet at each point is given by the function
δ(x, y) = 16 − x2 − y 2 . Find the mass of the metal sheet. (Hint: make
a sketch of D and use polar coordinates)
4. Find the volume ofpthe region bounded by the sphere x2 + y 2 + z 2 = 9
and the cone z = x2 + y 2 . (Hint: in spherical coordinates the cone
has equation φ = π4 )
5. Use the transformation T : (u, v) 7→ (x, y) where x = √u2 and y = √v3
to find the area bounded by the ellipse 2x2 + 3y 2 = 6. (Hint: First
find the preimage of the ellipse under T and compute the Jacobian
for T .)
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