Q. 1 Let [a1,a2],[b1,b2],[c1,c2]聽be ... [FREE SOLUTION]

Chapter 13: Q. 1 (page 1017)

Let $[a_{1}, a_{2}], [b_{1}, b_{2}], [c_{1}, c_{2}]$ be three closed intervals explain why the triple integral ${鈭�}_{a_{1}}^{a_{2}} {鈭�}_{b_{1}}^{b_{2}} {鈭�}_{c_{1}}^{c_{2}} d z d y d x$ 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

Expert verified

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 ${鈭�}_{a_{1}}^{a_{2}} {鈭�}_{b_{1}}^{b_{2}} {鈭�}_{c_{1}}^{c_{2}} d z d y d x$

Explaination

On evaluating the triple integral we get,

${鈭�}_{a_{1}}^{a_{2}} {鈭�}_{b_{1}}^{b_{2}} {鈭�}_{c_{1}}^{c_{2}} d z d y d x = {鈭�}_{a_{1}}^{a_{2}} {鈭�}_{b_{1}}^{b_{2}} [c_{2} - c_{1}] d y d x = {鈭�}_{a_{1}}^{a_{2}} [b_{2} - b_{1}] [c_{2} - c_{1}] d x = [a_{2} - a_{1}] [b_{2} - b_{1}] [c_{2} - c_{1}]$

We know the volume of a rectangular region with length width and height is $l w h$ . 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

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Let [a1,a2],[b1,b2],[c1,c2]be three closed intervals explain why the triple integral 鈭�a1a2鈭�b1b2鈭�c1c2dzdydxcomputes the volume of the rectangular solid with lengtha2-a1, widthb2-b1 and heightc2-c1

Short Answer

Step by step solution

Given information

Explaination

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Applied Mathematics

Calculus

Geometry

Probability and Statistics

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