Question

Let $\alpha ,\beta ,\gamma ,\delta ,ϵ,\text{and}\zeta$be real numbers, and let $\rho (x,y,z)$be a function giving the density at each point of a three-dimensional rectangular solid. What does the triple integral${\int }_{\alpha }^{\beta }{\int }_{\gamma }^{\delta }{\int }_{ϵ}^{\zeta }\rho (x,y,z)dzdydx$represent?

Short answer

The triple-iterated integral represents volume.

${\int }_{\alpha }^{\beta }{\int }_{\gamma }^{\delta }{\int }_{ϵ}^{\zeta }\rho (x,y,z)dzdydx={\iiint }_{\mathrm{ℝ}}\rho (x,y,z)dV$

Step-by-step solution

Step 1. Given information. 

The given integral is${\int }_{\alpha }^{\beta }{\int }_{\gamma }^{\delta }{\int }_{ϵ}^{\zeta }\rho (x,y,z)dzdydx.$

Step 2. Explanation. 

If $\alpha ,\beta ,\gamma ,\delta ,ϵ,\text{and}\zeta$are real numbers, and $\rho (x,y,z)$isa function giving the density at each point of a three-dimensional rectangular solid, then

$$\begin{gathered} {\int }_{\alpha }^{\beta }{\int }_{\gamma }^{\delta }{\int }_{ϵ}^{\zeta }\rho (x,y,z)dzdydx={\int }_{\alpha }^{\beta }\left({\int }_{\gamma }^{\delta }\left({\int }_{ϵ}^{\zeta }\rho (x,y,z)dz\right)dy\right)dx \\ {\int }_{\alpha }^{\beta }{\int }_{\gamma }^{\delta }{\int }_{ϵ}^{\zeta }\rho (x,y,z)dzdydx={\iiint }_{\mathrm{ℝ}}\rho (x,y,z)dV \end{gathered}$$

So the integral represents the volume.