MULTIVARIABLE INTEGRATION

MULTIVARIABLE INTEGRATION (SPHERICAL POLAR COORDINATES)

Question 1 a) Determine with the aid of a diagram an expression for the volume element in r, θ, ϕ. spherical polar coordinates, ( ) [You may not use Jacobians in this part] b) Use spherical polar coordinates to obtain the standard formula for the volume of a sphere of radius a. dv = r 2 sinθ dr dθ dϕ

Question 2 Use spherical polar coordinates, ( r, θ, ϕ ), to obtain the standard formula for the surface area of a sphere of radius a. 2 2 da = a sinθ dθ dϕ A = 4π a

Question 3 Determine an exact simplified value for 1 dx dy dz, R ( x 2 + y 2 + z 2 ) 2 where R is the region outside the sphere with equation x + y + z = 1. 4π

Question 4 The finite R region is defined as 1 x + y + z 4. Determine an exact simplified value for 1 dx dy dz. R ( ) 3 2 x + y + z 4π ln 2

Question 5 Determine an exact simplified value for 2 15z dx dy dz, R where R is the region between the spheres with equations x + y + z = 1 and x + y + z = 4. 124π

Question 6 The finite R region is defined as the interior of the sphere with equation x + y + z = 1. Determine an exact simplified value for ( ) 2 2 x + y + z 2 e x y z dx dy dz. R 2 2π 1 e

Question 7 A hemispherical solid piece of glass, of radius a m, has small air bubbles within its volume. The air bubble density ρ ( z), in 3 m, is given by ( z) ρ = k z, where k is a positive constant, and z is a standard cartesian coordinate, whose origin is at the centre of the flat face of the solid. Given that the solid is contained in the part of space for which z 0, determine the total number of air bubbles in the solid. MM2B, 1 4 ka 4 π

Question 8 A uniform solid has equation 2 x + y + z = a, with x > 0, y > 0, z > 0, a > 0. Use integration in spherical polar coordinates, ( r, θ, ϕ ), to find in Cartesian form the coordinates of the centre of mass of the solid. 3 ( a, 3 a, 3 a ) 8 8 8

Question 9 A solid sphere has equation 2 x + y + z = a. The density, ρ, at the point of the sphere with coordinates (,, ) x y z is given by 1 1 1 2 2 ρ = x + y. 1 2 Determine the average density of the sphere. MM2A, ρ = 3 π a 16

Question 10 A thin uniform spherical shell with equation 2 x + y + z = a, a > 0, occupies the region in the first octant. Use integration in spherical polar coordinates, ( r, θ, ϕ ), to find in Cartesian form the coordinates of the centre of mass of the shell. a (, a, a )

Question 11 The finite region Ω is defined as x + y + z 1. Use Spherical Polar Coordinates ( r, θ, ϕ ), to evaluate the volume integral ( r sinθ cosϕ ) 4 dv. Ω 4π 35

Question 12 A solid sphere has equation x + y + z = 1. The region defined by the double cone with Cartesian equation 3z x + y, is bored out of the sphere. Determine the volume of the remaining solid. MM2C, V = 2π 3

Question 13 A solid sphere has radius 5 and is centred at the Cartesian origin O. The density ρ at point (,, ) P x y z of the sphere satisfies 1 1 1 ρ = 3 1 + z 85 1 x1 + y1 + z 1. Use spherical polar coordinates, ( r, θ, ϕ ), to find the mass of the sphere. m = 50π

Question 14 A solid uniform sphere has mass M and radius a. Use spherical polar coordinates, ( r, θ, ϕ ), to show that the moment of inertial of this sphere about one of its diameters is 2 2 5 Ma. proof

Question 15 A thin uniform spherical shell has mass m and radius a. Use spherical polar coordinates, ( r, θ, ϕ ), to show that the moment of inertial of this spherical shell about one of its diameters is 2 2 3 ma. proof

Question 16 The finite region R is defined as the region enclosed by the ellipsoid with Cartesian equation x y z + + = 1. 9 16 25 By first transforming the Cartesian coordinates into a new Cartesian coordinate r, θ, ϕ, find the value of system, use spherical polar coordinates, ( ) ( x + y + z ) dx dy dz. R 800π

Question 17 A solid uniform sphere of radius a, has variable density ( r) radial distance of a given point from the centre of the sphere. ρ = r, where r is the a) Use spherical polar coordinates, ( r, θ, ϕ ), to find the moment of inertia of this sphere I, about one of its diameters. b) Given that the total mass of the sphere is m, show that I 4 2 = ma. 9 I = 4 πa 9 6

Question 18 Evaluate the triple integral 2 5x dxdydz, V where V is the finite region contained within the closed surface with equation 2 2 2 y z x + + = 1. 4 9 8π

Question 19 The finite R region is defined as x + y + z 2z. Determine an exact simplified value for z x + y + z dx dy dz. R 16π 15

Question 20 The finite R region is defined as 4z x + y + z 16z. Determine an exact simplified value for 3 z dx dy dz 8. R MM2E, 1092π

Question 21 A solid sphere has equation 2 x + y + z = a, a > 0. The sphere has variable density ρ, given by ( z) ρ = k a, k > 0. Use integration in spherical polar coordinates, ( r, θ, ϕ ), to find in Cartesian form the coordinates of the centre of mass of the sphere. ( 0,0, 1 ) 5 a

Question 22 A solid is defined in a Cartesian system of coordinates by 2 2 x + y = xz, 0 z 2. a) Describe the solid with the aid of a sketch. b) Use standard elementary formulas to find the volume of the solid. c) Use spherical polar coordinates to verify the answer to part (b) V = 2π 3

Question 23 A solid sphere has radius a and mass m. The density ρ at any point in the sphere is inversely proportional to the distance of this point from the centre of the sphere Show that the moment of inertia of this sphere about one of its diameters is 2 1 3 ma proof

Question 24 The finite R region is defined as 1 x + y + z 2z. Determine an exact simplified value for z dx dy dz. R 9π 8

Question 25 The finite R region is defined as x + y + z 4z and z 2. Determine an exact simplified value for 2 z x + y + z dx dy dz. R 44π 3

Question 26 A non right circular cone has Cartesian equation 2 2 x + y = xz, 0 z 2. Use spherical polar coordinates to find an exact simplified value for xz dv, R where R is the interior of the cone. 4π 5

Question 27 A solid uniform sphere has mass M and radius a. Use spherical polar coordinates, ( r, θ, ϕ ), and direct calculus methods, to show that the moment of inertial of this sphere about one of its tangents is 7 2 Ma. 5 You may not use any standard rules or standard results about moments of inertia in this question apart from the definition of moment of inertia. proof