
Successive Partial Integration 16-31
= − − + +
∫
= − − + − + −
−
6 9 3
18 9 27
27
2
9 9
2 2 2
0
3
3
2
0
2
2
y xy y x y xy dx
x x
x
x
x
99
4
9
9
2
9 9
9
4
9 18
45
4
9
2
2
3
2
3
0
2
2
x
x
x
x x
x
dx
x
x
+ − + − +
∫
= − + +
xx
dx
x x
x x
3
0
2
2
3 4
0
2
4
9 9
15
4
9
16
3
∫
= − + −
= ..
Example 16.31 Change the order of integration in the triple integral
( )1
0
6 3 2
0
3
3
2
0
2
−
∫∫∫
− −
−
x dz dy dx
x y
x
that we worked out in the previous example so that dy dx dz appears.
Solution: Unlike in the previous example, instead of z, we put y = 0 and solve the equation of the
plane 3x + 2y + z - 6 = 0 for y to obtain
y
x z
= − −3
3
.
The lower limit of x is 0.
Setting y = 0, solving for x