Question

Let $[a_1,a_2],[b_1,b_2],[c_1,c_2]$be three closed intervals explain why the triple integral ${\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}dzdydx$computes the volume of the rectangular solid with length$a_2-a_1$, width$b_2-b_1$ and height$c_2-c_1$

Short answer

On evaluating the triple integral we get a value which is equal to the volume of the rectangular box with given length width and height

Step-by-step solution

Given information 

We are given a triple integral${\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}dzdydx$

Explaination

On evaluating the triple integral we get,

$$\begin{gathered} {\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}dzdydx \\ ={\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}[c_2-c_1]dydx \\ ={\int }_{a_1}^{a_2}[b_2-b_1][c_2-c_1]dx \\ =[a_2-a_1][b_2-b_1][c_2-c_1] \end{gathered}$$

We know the volume of a rectangular region with length width and height is $lwh$. Hence this integral represents the volume of the rectangular region with length width and height$[a_2-a_1][b_2-b_1][c_2-c_1]$ respectively