Question

Problem Zero: Read the section and make your own summary of the material.

Short answer

If $a_1<a_2,b_1<b_2,\text{and}{c}_1<c_2$be real numbers, let R be the rectangular solid defined by $\mathcal{R}=\left\{\right(x,y,z)\mid a_1\le x\le a_2,b_1\le y\le b_2,\text{and}{c}_1\le z\le c_2\}$and $f(x,y,z)$is a continuous function defined on R then,

Riemann Sum is defined as$\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i^{\ast },y_j^{\ast },z_k^{\ast })\mathrm{\Delta }V$

and triple integral of f over Ris ${\iiint }_{\mathcal{R}}f(x,y,z)dV=\underset{\mathrm{\Delta }\to 0}{lim}\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i^{\ast },y_j^{\ast },z_k^{\ast })\mathrm{\Delta }V$

iterated triple integral is

$$\begin{gathered} {\iiint }_{\mathrm{ℝ}}f(x,y,z)dV={\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}f(x,y,z)dzdydx \\ ={\int }_{a_1}^{a_2}\left({\int }_{b_1}^{b_2}\left({\int }_{c_1}^{c_2}f\right(x,y,z\left)dz\right)dy\right)dx \end{gathered}$$

Step-by-step solution

Step 1. Given information

The given topic of the section is Triple Integrals.

Step 2. Summary

If $a_1<a_2,b_1<b_2,\text{and}{c}_1<c_2$be real numbers, let R be the rectangular solid defined by $\mathcal{R}=\left\{\right(x,y,z)\mid a_1\le x\le a_2,b_1\le y\le b_2,\text{and}{c}_1\le z\le c_2\}$and $f(x,y,z)$is a continuous function defined on R then,

Riemann Sum is defined as $\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i^{\ast },y_j^{\ast },z_k^{\ast })\mathrm{\Delta }V$

and triple integral of f over Ris ${\iiint }_{\mathcal{R}}f(x,y,z)dV=\underset{\mathrm{\Delta }\to 0}{lim}\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i^{\ast },y_j^{\ast },z_k^{\ast })\mathrm{\Delta }V$

iterated triple integral is

$$\begin{gathered} {\iiint }_{\mathrm{ℝ}}f(x,y,z)dV={\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}f(x,y,z)dzdydx \\ ={\int }_{a_1}^{a_2}\left({\int }_{b_1}^{b_2}\left({\int }_{c_1}^{c_2}f\right(x,y,z\left)dz\right)dy\right)dx \end{gathered}$$