Question

The triple integral is used to compute
1. Volume
2. Area
3. Both volume and area
4. None of the above

Answer 1

Correct Answer - Option 1 : Volume

Concept:

Double integral:

Let f(x, y) be defined at each point in a region ‘R’. Let the region ‘R’ be divided into ‘n’ sub-regions each of area δA₁, δA₂, …….. δA_n

Let (x_i – y_i) be an arbitrary point in a sub-region with area δA_i

Then,

\(\begin{array}{*{20}{c}} {lim}\\ {n \to \infty } \end{array}\left[ {\mathop \sum \limits_{{\rm{i}} = 1}^{\rm{n}} {\rm{f}}\left( {{{\rm{x}}_{\rm{i}}},{{\rm{y}}_{\rm{i}}}} \right){\rm{\delta }}{{\rm{A}}_{\rm{i}}}} \right] = \int\!\!\!\int {\rm{f}}\left( {{\rm{x}},{\rm{\;y}}} \right){\rm{dxdy}}\)

Triple integral:

Let f(x, y, z) be a function defined over a 3-dimensional finite region V. Divide the region V into elementary volumes δV₁, δV₂, …….. δV_n

Let (x_r, y_r, z_r) be any point in the r^th sub-division δV_r

Then,

\(\begin{matrix} lim \\ n\to \infty \\ \end{matrix}\left[ \underset{\text{r }\!\!~\!\!\text{ }=1}{\overset{\text{n}}{\mathop \sum }}\,\text{f}\left( {{\text{x}}_{\text{r}}},{{\text{y}}_{\text{r}}},\text{ }\!\!~\!\!\text{ }{{\text{z}}_{\text{r}}} \right)\text{ }\!\!\delta\!\!\text{ }{{\text{V}}_{\text{r}}} \right]=\iiint{\text{f}\left( \text{x},\text{ }\!\!~\!\!\text{ y},\text{ }\!\!~\!\!\text{ z} \right)\text{dV}}\)

Hence the double integrals are used for computing area and triple integrals are used for computing volume.