CHAPTER 7 DOUBLE AND TRIPLE INTEGRALS EXERCISE 78 Page 755. Evaluate: dxd y. is integrated with respect to x between x = and x =, with y regarded as a constant dx= [ x] = [ 8 ] = [ ] ( ) ( ) d x d y = d y y () = = = d x d y =. Evaluate: ( x y )d x d y. (x y) is integrated with respect to x between x = and x =, with y regarded as a constant ( x y)d x = ( y) x = [ x xy] = ( () y) ( () y) = (9 y) ( y) = 9 y +y = 5 y y () () ( x y) d xd y = (5 y) d y = 5y 5() 5() = = ( ) (5.5) = 5 +.5 =.5 ( x y )d x d y =.5. Evaluate: ( x y )d y d x (x y) is integrated with respect to y between x = and x =, with x regarded as a constant, John Bird
y ( x y)d y= ( xy ) = ( x)() ( x)() = ( 6x.5) ( x ) [ ] ( ) ( ) ( x y ) d y d x= ( x.5) d x= x.5 x = 5.5 = x.5 = 5 +.5 =.5 ( x y )d y d x=.5. Evaluate: 5. ( x 5 y)dxd y (x 5y) is integrated with respect to x between x = and x =, with y regarded as a constant ( x 5 y)d x= (5 yx ) = [( y) (.5 + 5 y) ] 5 5 = y.5 5y =.5 5y 5y ( x 5 y) d xd y = (.5 5 y) d y =.5y = ( 7.5 87.5) (.5 7.5) 5 = 7.5 87.5.5 + 7.5 = 7 5 ( x 5 y )d x d y = 7 5. Evaluate: 6 5 ( + )d d x y x y. ( x y) + is integrated with respect to x between x = and x = 5 5 5 5 8 ( x + y)d x= ( yx ) y 8y + = + + = 5 + y 8 8y = 9 + y 6 5 6 6 [ ] ( ) ( ) ( x + y ) d x d y = (9 + y ) d y = 9 y+ 6 y = + 6 9 + 6, John Bird
= + 6 9 6 = 5 6 5 ( x + y )d x d y = 5 6. Evaluate: ( )d d xy x y ( xy ) is integrated with respect to x between x = and x = ( ) ( ) ( ) ( xy ) d x = y = 6y.5y = 7.5y 7.5y ( xy ) d x d y = ( 7.5y ) d y = = ( 6) (.5) = 6 +.5 = 57.5 ( xy )d x d y = 57.5 7. Evaluate: (+ sin θ)dθdr. ( + sin θ ) is integrated with respect to θ between θ = and θ =, ( + sin θ)d x = θ cos θ cos cos = = (.5) (.5) = ( + sin θ )d θ d r = ( ) d r = [ r] = ( 9 ) ( ) = 9 + = or 7.7 ( + sin θ)dθd r = or 7.7 8. Evaluate: d x ( xy )d y. With this configuration: ( xy) is integrated with respect to y between x = and x =, John Bird
( xy) d y = [ y xy ] = ( 6 6x) ( 8 x) = 6 6x 8 + x = 8 x d ( ) d = d ( 8 ) = ( 8 ) d = [ 8 6 ] = ( 5 ) ( 8 6 ) x xy y x x x x x x = 5 8 + 6 = d x ( xy)d y = 9. The volume of a solid V, bounded by the curve x y between the limits x = to x = and y = to y = is given by: Evaluate V. V = ( x y )d x d y ( x y) is integrated with respect to x between x = and x =, ( x y)d x = x yx = (.5 y) ( ) x =.5 y y ( x y) d xd y = (.5 y) d y =.5y = ( 7 ) ( ) = 5 volume of solid, V = ( x y )d x d y = 5. The second moment of area I of a 5 cm by cm rectangle about an axis through one corner perpendicular to the plane of the figure is given by: Evaluate I. 5 ( )d d I = x + y y x ( x y ) + is integrated with respect to y between y = and y =, ( x + y)d y= xy + = ( x + 9) ( ) y = x + 9, John Bird
5 5 5 ( x + y )d y d x= ( x 9 ) d x= [ x + 9 x] = ( 5 + 5 ) ( ) = 7 second moment of area, 5 I = ( x + y )d y d x= 7cm, John Bird
EXERCISE 79 Page 756. Evaluate:. (8 xyz) d z d x d y (8xyz) is integrated with respect to z between z = and z =, with x and y regarded as constants (8 xyz) d z = [ xyz ] = ( xy) ( ) = xy xy is integrated with respect to x between x = and x =, with y regarded as a constant ( xy)d x = [ x y] = ( 8y) ( 8y) = y y is integrated with respect to y between y = and y = [ ] ( ) ( ) ( y) d y = 5y = 5 = 5 (8 xyz ) d z d x d y = 5. Evaluate:. (8 xyz) d x d y d z (8xyz) is integrated with respect to x between x = and x =, with y and z regarded as constants (8 xyz) d x = [ x yz] = ( yz) ( ) = yz yz is integrated with respect to y between y = and y =, with z regarded as a constant ( yz)d y = [ y z] = ( 8z) ( 8z) = z z is integrated with respect to z between z = and z = [ ] ( ) ( ) ( z) d z = 5z = 5 = 5 (8 xyz ) d x d y d z = 5 5, John Bird
. Evaluate: ( + + ) x y z dxd ydz. ( x y z) + + is integrated with respect to x between x = and x =, with y and z regarded as constants, ( x+ y + z)d a = + ( y) x+ ( z) x = (.5 + y + z ) (.5 + y + z ) = + y + z ( + y + z ) is integrated with respect to y between y = and y =, with z regarded as a constant, y 6 + y + z d y = y+ + ( z ) y 8 z z = + + ( ) = 8 + 6z ( 8 + 6z ) is integrated with respect to z between z = and z = ( 8 + 6z ) d z = 8z+ = ( 6 + ) ( ) 6z ( ) = 6 x+ y + z d xd yd z = 6. Evaluate: ( + 5 )d d d x y z x y z. ( x 5y ) z + is integrated with respect to x between x = and x =, with y and z regarded as constants, x 8 ( x + 5y )d z x= + (5 y ) x () z x y z ( ) = + 8 + y z = 6, John Bird
8 + y z) is integrated with respect to y between y = and y =, with z regarded as a ( constant, 8 8 y 8 8 + y z d y = y ( z) y z z + = + + = 8z ( 8 z) is integrated with respect to z between z = and z = ( ) [ ] ( ) ( ) 8z d z = z z = 6 6 6 = 8 ( x + 5y z)dxd ydz = 8 5. Evaluate: ( xy sin z )d x d y d z. (xy sin z) is integrated with respect to x between x = and x =, with y and z regarded as constants, ( xysin z)d x= ysin z = ysin z ( ) = sin y z ( sin y z ) is integrated with respect to y between y = and y =, with z regarded as a constant, y ysin z d y sin z sin z ( ) = = = sin z sin z is integrated with respect to z between z = and z = 7, John Bird
sin z d z cos z = = = ( xy sin z)d x d y d z = or 8.7 6. Evaluate:. ( xy)dxd y dz (xy) is integrated with respect to x between x = and x =, with y regarded as constants, ( ) ( ) ( xy)d x = y = y.5y =.5y.5y is integrated with respect to y between y = and y =, ( y) y ( ) ( ).5 y.5 d = =.75 =.5.5 is integrated with respect to z between z = and z = ( ) [ ] ( ) ( ).5 d z =.5z = 9 = 9 ( xy)dxd y dz = 9 7. Evaluate:. ( xz + y)dxd y dz (xz + y) is integrated with respect to x between x = and x =, with y and z regarded as constants, ( ) ( ) ( xz+ y)d x= z+ yx =.5z+ y.5z y = y y is integrated with respect to y between y = and y = ( y) y ( y ) ( ) ( ) d = = = is integrated with respect to z between z = and z = 8, John Bird
( ) [ ] ( ) ( ) d z = z = = 8 ( xz + y)dxd y dz = 8 8. A box shape X is described by the triple integral: Evaluate X. X = ( x+ y+ z)dzd ydx (x + y + z) is integrated with respect to x between x = and x =, with y and z regarded as constants ( ) ( ) ( x+ y+ z)d x= + yx+ zx =.5 + y+ z =.5 + y+ z.5 + y + z is integrated with respect to y between y = and y = ( ) ( ) ( ) y.5 + y + z d y =.5y + + zy = + + z = + z + z is integrated with respect to z between z = and z = ( ) [ ] ( ) ( ) + z dz = z+ z = 9+ 9 = 8 X = ( x+ y+ z )d z d y d x= 8 9, John Bird