MULTIVARIABLE INTEGRATION

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1 MULTIVARIABLE INTEGRATION (PLANE & CYLINDRICAL POLAR COORDINATES)

2 PLANE POLAR COORDINATES

3 Question 1 The finite region on the x-y plane satisfies 1 x + y 4, y 0. Find, in terms of π, the value of I. I = x x + y dx dy. R 7 6 π

4 Question The finite region on the x-y plane satisfies 1 x + y 16, y 0. Find, in terms of π, the value of I. ( ). I = x + y dx dy R 55π

5 Question 3 Find the exact simplified value for the following integral. ( x y ) y + e dx dy. 0 0 π 8 Question 4 Find the exact simplified value for the following integral. ( ) 5 3 ( x y ) e x + + y dx dy. 0 0 π 10

6 Question 5 The finite region on the x-y plane satisfies 4 x + y 4x, y 0. Find the value of I. I = xy dx dy. R 9

7 Question 6 The points A and B have Cartesian coordinates ( 1,0 ) and ( 1,1 ), respectively. The finite region R is defined as the triangle OAB, where O is the origin. Use plane polar coordinates, ( r, θ ) to determine the value of 3 x dx dy. x + y R [No credit will be given for workings in other coordinate systems.] π 1

8 Question 7 The finite region R is defined as 1 x + y 4. Determine an exact simplified value for ( x + y ) ln dx dy. x + y R 4π ( 1+ ln )

9 Question 8 I e x 0 = dx and I e y 0 = dy. By considering an expression for clearly that I and the use of plane polar coordinates, show I = 1 π. proof

10 Question 9 e x y dx dy. a) Use plane polar coordinates ( r, θ ), to find the exact simplified value of the above integral. b) Hence evaluate e x dx. e x y dx dy = π, e x dx = π

11 Question 10 The points A and B have Cartesian coordinates ( 0,1 ) and ( 1,1 ), respectively. The finite region R is defined as the triangle OAB, where O is the origin. Use plane polar coordinates, ( r, θ ) to determine the value of y dx dy. R [No credit will be given for workings in other coordinate systems.] 1 4

12 Question 11 The finite region R, on the x-y plane, satisfies x + y 1. Find, in terms of π, the value of ( y 3 y ) dx dy. R 3π 4

13 Question 1 Find the exact simplified value for the following integral. 0 0 dy dx. 1+ x + y 1 1 x π ( 1 ln )

14 Question 13 Find the exact simplified value for the following integral. 4 x x + y dy dx. 0 0 π

15 Question 14 The finite region R is bounded by the straight lines and curves with the following equations. y = 0, x = 0, x + y = 4 and y = 3x. Determine an exact simplified value for x x + y dx dy. R 3

16 Question 15 A uniform circular lamina of mass M and radius a. Use double integration to find the moment of inertia of the lamina, when the axis of rotation is a diameter. 1 4 Ma

17 Question 16 A circular sector of radius r subtends an angle of α at its centre O. The position of the centre of mass of this sector lies at the point G, along its axis of symmetry. Use calculus to show that OG r sinα =. 3α proof

18 Question 17 The finite region on the x-y plane satisfies 4 4 4x + 4y π 8x y and 6x + 6y π. Find the value of the following integral. cos( x + y ) dx dy. R MMC, 1 π

19 Question 18 The finite region R, on the x-y plane, satisfies x + y 1. Find, in terms of π, the value of I. ( ). I = + xy + x yx dx dy R π

20 Question 19 The finite region R is bounded by the straight lines with the following equations. x = 0, y = 0 and y = 1 x. Use plane polar coordinates, ( r, θ ) to determine the value of x + y dx dy. x + y R [No credit will be given for workings in other coordinate systems.] π

21 Question 0 I = 4 y y dx dy. y y x + y 0 Use polar coordinates to find an exact simplified answer for I. MM-D, 4 π

22 Question 1 The finite region R is bounded by the straight lines with the following equations. y = 0, x = 1, x = 1 and y = x. Use plane polar coordinates, ( r, θ ) to determine the value of x dx dy. R ( x + y ) [No credit will be given for workings in other coordinate systems.] ( ) 1 8 π +

23 Question The finite region R is defined as 4x x + y 8x. Determine the value of y dx dy. R 60π

24 Question 3 Use plane polar coordinates, ( r, θ ) to determine the value of e x dxdy. x= y x y= 0 MME, 1

25 Question 4 Use plane polar coordinates, ( r, θ ) to determine the value of e ( x + y ) dx dy, R where R is the region in the first quadrant in a standard Cartesian coordinate system. 1

26 Question 5 Given that µ is a positive constant determine the value of µ e ( x + y) dx dy, µ

27 Question 6 Determine an exact simplified value for ( ) ( x + y ) e x + y dx dy, R where R is the region x + y > 1 π e

28 Question 7 Find the exact simplified value for the following integral x + 16y dx dy. x + y y MM-B, π ( 4 π )

29 Question 8 The finite region R is bounded by the straight lines with the following equations. x + y = 1, x + y =, y = x and y = 0. Use plane polar coordinates ( r, θ ) to find the value of R ( + y) y x dx dy. 3 x [No credit will be given for workings in other coordinate systems.] 3 4

30 Question 9 The region R on the x-y plane is defined by the inequalities 1 x + y 5 and 1 x y x. Show clearly that ( x + y) dx dy = ( 5 5 ). 15 R proof

31 CYLINDRICAL COORDINATES

32 Question 1 Find the value of rz dv, Ω where Ω is the region inside the cylinder with equation x + y = 4, z In this question use cylindrical polar coordinates ( r,, z) θ. 56π 9

33 Question Find the value of ( x + y + z) dx dydz, where V is the region inside the cylinder with equation V x + y = 1, 0 z 4. 4 π

34 Question 3 Find in exact form the volume enclosed by the cylinder with equation x + y = 16, z 0, and the plane with equation z = 1 x. 19π

35 Question 4 Find the volume of the region bounded by the cylinder with equation x + y = 4, and the surfaces with equations z = x + y and z = 0. 8π

36 Question 5 Find the volume of the paraboloid with equation z = 1 x y, z 0. π

37 Question 6 The finite region Ω is enclosed by the cylinder with Cartesian equation x + y = 1, 1 z 1. Determine an exact simplified value for 3 ( 1 ) e x + + z y dx dy dz. Ω π [ e 1]

38 Question 7 The finite region V is enclosed by the cone with Cartesian equation z = x + y, 0 z 6. Determine an exact simplified value for x + y + z dx dy dz. V 16 π

39 Question 8 The height z, of a cooling tower, is 10 m. The radius r m, of any of the circular cross sections of the cooling tower is given by the equation Use cylindrical polar coordinates ( r, θ, z) 1 ( z ) r = , to find the volume of the tower π

40 Question 9 Use cylindrical polar coordinates ( r,, z) θ to find the volume of the region defined as ( ) x + y + z + 4 5, z π

41 Question 10 Find the value of ( 1+ xy) dv, V where V is the finite region enclosed by the surface with Cartesian equation z = 1 x y, z 0. π

42 Question 11 Find in exact form the volume of the solid defined by the inequalities x + y 4, x 0, y 0 and 0 z 6 xy. 6π Question 1 Find the volume of the finite region bounded by the surfaces with Cartesian equations z = 13 4x 4y and z = 1 x y V = 4π

43 Question 13 A scalar field F exist inside the cylinder with equation x + y = 1, 0 z 4. Given further that (,, ) 3 F x y z + xy + yz, evaluate the integral, V F dv where V denotes the region enclosed by the cylinder 14π

44 Question 14 Use cylindrical polar coordinates ( r,, z) θ to evaluate 5yz x + y dxdydz, V where V is the region defined as x + y y, contained within the sphere with equation x + y + z =

45 Question 15 The finite region Ω is defined by the inequalities x + y 1 and z x + y. Use cylindrical polar coordinates to evaluate 6x dx dy dz. Ω 1π 5

46 Question 16 The finite region V is defined by the inequalities x + y + z 1 and z 1 x + y. Use cylindrical polar coordinates to evaluate z dx dy dz. V π 1

47 Question 17 Use cylindrical polar coordinates ( r,, z) cone of height h and base radius a is θ to show that the volume of a right circular 1 π a h. 3 proof

48 Question 18 a) Determine with the aid of a diagram or a Jacobian matrix an expression for the r, θ. area element in plane polar coordinates, ( ) A cylinder of radius 1 a is cut out of a sphere of radius a. b) Find a simplified expression for the volume of the cylinder, given that one of its generators passes through the centre of the sphere dxdy = r dr dθ, V = [ 3 π 4 9 ]

49 Question 19 The region V is contained by the paraboloid with Cartesian equation y = x + z, 0 y 4. Determine an exact simplified value for x + z dx dydz. V π

50 Question 0 Use cylindrical polar coordinates, ( r, θ, z) with Cartesian equation, to find the exact volume of the ellipsoid x + y + 3z = 1. 4π V = 3 3

51 Question 1 The finite region V is bounded by surfaces with Cartesian equations ( ) 4 z = 4 x + y, z 0 and x + y + z = 3, z 0. Use cylindrical polar coordinates ( r,, z) π 15 θ to show that the volume of V is ( ). proof

52 Question Use cylindrical polar coordinates, ( r, θ, z) defined by the following Cartesian inequalities, to find the exact volume of the region z x + y, x + y 1 and z 6. 00π V = 3

53 Question 3 Use cylindrical polar coordinates, ( r, θ, z) by the following Cartesian inequalities, to find the volume of the region defined z 4 x y, z 4 + x + y and x + y 4. V = 16π

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