Answer
(a) equal
(b) equal
(c) not equal
Work Step by Step
We have the triple integral: $\mathop \smallint \limits_0^1 \mathop \smallint \limits_3^4 \mathop \smallint \limits_6^7 f\left( {x,y,z} \right){\rm{d}}z{\rm{d}}y{\rm{d}}x$.
From the order of the integral we obtain the domain:
(1) ${\ \ \ \ \ }$ $0 \le x \le 1$, ${\ \ }$ $3 \le y \le 4$, ${\ \ }$ $6 \le z \le 7$
Now, we compare with the following triple integrals:
(a) $\mathop \smallint \limits_6^7 \mathop \smallint \limits_0^1 \mathop \smallint \limits_3^4 f\left( {x,y,z} \right){\rm{d}}y{\rm{d}}x{\rm{d}}z$
From the order of the integral we obtain the domain:
$6 \le z \le 7$, ${\ \ \ }$ $0 \le x \le 1$, ${\ \ \ }$ $3 \le y \le 4$
Since the domain is the same with (1), the triple integral is equal to $\mathop \smallint \limits_0^1 \mathop \smallint \limits_3^4 \mathop \smallint \limits_6^7 f\left( {x,y,z} \right){\rm{d}}z{\rm{d}}y{\rm{d}}x$.
(b) $\mathop \smallint \limits_3^4 \mathop \smallint \limits_0^1 \mathop \smallint \limits_6^7 f\left( {x,y,z} \right){\rm{d}}z{\rm{d}}x{\rm{d}}y$
From the order of the integral we obtain the domain:
$3 \le y \le 4$, ${\ \ \ }$ $0 \le x \le 1$, ${\ \ \ }$ $6 \le z \le 7$
Since the domain is the same with (1), the triple integral is equal to $\mathop \smallint \limits_0^1 \mathop \smallint \limits_3^4 \mathop \smallint \limits_6^7 f\left( {x,y,z} \right){\rm{d}}z{\rm{d}}y{\rm{d}}x$.
(c) $\mathop \smallint \limits_0^1 \mathop \smallint \limits_3^4 \mathop \smallint \limits_6^7 f\left( {x,y,z} \right){\rm{d}}x{\rm{d}}z{\rm{d}}y$
From the order of the integral we obtain the domain:
$0 \le y \le 1$, ${\ \ \ }$ $3 \le z \le 4$, ${\ \ \ }$ $6 \le x \le 7$
The domain is not the same with (1), therefore the triple integral is not equal to $\mathop \smallint \limits_0^1 \mathop \smallint \limits_3^4 \mathop \smallint \limits_6^7 f\left( {x,y,z} \right){\rm{d}}z{\rm{d}}y{\rm{d}}x$.
