Q. 1 - Calculus

Question

Let be real $\alpha ,\beta ,\gamma ,\delta ,ϵ,\text{and }\zeta$numbers. Evaluate the triple iterated integral

${\int }_{\alpha }^{\beta }{\int }_{\gamma }^{\delta }{\int }_{ϵ}^{\zeta }dzdydx$

What does this integral represent?

Step-by-Step Solution

Answer

Integral is evaluated as ${\int }_{\alpha }^{\beta }{\int }_{\gamma }^{\delta }{\int }_{ϵ}^{\zeta }dzdydx=\left(\zeta -ϵ\right)\left(\delta -\gamma \right)\left(\beta -\alpha \right)$and represents volume.

Step 1. Given information

The given integral is ${\int }_{\alpha }^{\beta }{\int }_{\gamma }^{\delta }{\int }_{ϵ}^{\zeta }dzdydx$

Step 2. Simplification

Evaluate the integral

$$\begin{gathered} {\int }_{\alpha }^{\beta }{\int }_{\gamma }^{\delta }{\int }_{ϵ}^{\zeta }dzdydx={\int }_{\alpha }^{\beta }\left({\int }_{\gamma }^{\delta }\left({\int }_{ϵ}^{\zeta }1 dz\right)dy\right)dx \\ {\int }_{\alpha }^{\beta }\left({\int }_{\gamma }^{\delta }\left({\left[z\right]}_{ϵ}^{\zeta }\right)dy\right)dx \\ ={\int }_{\alpha }^{\beta }\left({\int }_{\gamma }^{\delta }\left(\zeta -ϵ\right)dy\right)dx \\ ={\int }_{\alpha }^{\beta }\left({\left[\zeta -ϵ\right]}_{\gamma }^{\delta }\right)dx \\ ={\int }_{\alpha }^{\beta }\left(\left(\zeta -ϵ\right)\left(\delta -\gamma \right)\right)dx \\ ={\left[\left(\zeta -ϵ\right)\left(\delta -\gamma \right)\right]}_{\alpha }^{\beta } \\ =\left(\zeta -ϵ\right)\left(\delta -\gamma \right)\left(\beta -\alpha \right) \end{gathered}$$

So integral is evaluated as ${\int }_{\alpha }^{\beta }{\int }_{\gamma }^{\delta }{\int }_{ϵ}^{\zeta }dzdydx=\left(\zeta -ϵ\right)\left(\delta -\gamma \right)\left(\beta -\alpha \right).$

The triple-iterated integral represents volume.