Question 1. Using a double integral, ﬁnd the area of the “boomerang”-shaped subset of R2 lying between the two parabolas: x = 4−y2 and x = 8−2y2.

Question 2. A hemispherical bowl of radius 10 cm is ﬁlled with liquid to a depth of 6 cm.

(a) Find the exact volume of liquid in the bowl using these hints: • Visualize the bowl sitting on the origin, so that it is part of the sphere: x2 + y2 + (z−10)2 = 100. • Sketch the bowl, and mark on it the depth of liquid. • Use cylindrical coordinates, because the variable z is distinguished. The possible values for z and θ are straightforward, and the possible values for ρ can be expressed in terms of z using the sphere equation.

(b) The liquid in the bowl is a suspension of medicine. As it sits, the active ingredient settles towards the bottom of the bowl. At some later time, the density of active ingredient (in micrograms per cm3) is described (in cylindrical coordinates) by the function

f(ρ,θ,z) = 100−

5 2

z2.

Find the total mass of active ingredient in the bowl (exactly).

2018-03-05
Maths