Chapter 13: Q. 71 (page 1057)

$L e t R = \{(x, y, z) | a_{1} ⩽ x ⩽ a_{2}, b_{1} ⩽ y ⩽ b_{2}, c_{1} ⩽ z ⩽ c_{2}\} . P r o v e t h a t : ∭_{R} d V = (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) . W h a t i s t h e r e l a t i o n s h i p b e t w e e n R a n d t h e p r o d u c t (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) .$

Short Answer

Expert verified

$∭_{R} d V = \int_{a_{1}}^{a_{2}} \int_{b_{1}}^{b_{2}} \int_{c_{1}}^{c_{2}} d z d y d x = \int_{a_{1}}^{a_{2}} \int_{b_{1}}^{b_{2}} [\int_{c_{1}}^{c_{2}} d z] d y d x = (c_{2} - c_{1}) \int_{a_{1}}^{a_{2}} [\int_{b_{1}}^{b_{2}} d y] d x = (c_{2} - c_{1}) (b_{2} - b_{1}) \int_{a_{1}}^{a_{2}} d x = (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) ∭_{R} d V = (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) .$

Step by step solution

Step 1. Given Information.

$G i v e n : R = \{(x, y, z) | a_{1} ⩽ x ⩽ a_{2}, b_{1} ⩽ y ⩽ b_{2}, c_{1} ⩽ z ⩽ c_{2}\}$

Step 2. Proof.

$T h e t r i p l e i n t e g r a l i s e v a l u a t e d a s f o l l o w s : ∭_{R} d V = \int_{a_{1}}^{a_{2}} \int_{b_{1}}^{b_{2}} \int_{c_{1}}^{c_{2}} d z d y d x = \int_{a_{1}}^{a_{2}} \int_{b_{1}}^{b_{2}} [\int_{c_{1}}^{c_{2}} d z] d y d x = \int_{a_{1}}^{a_{2}} \int_{b_{1}}^{b_{2}} (c_{2} - c_{1}) d y d x = (c_{2} - c_{1}) \int_{a_{1}}^{a_{2}} [\int_{b_{1}}^{b_{2}} d y] d x = (c_{2} - c_{1}) \int_{a_{1}}^{a_{2}} (b_{2} - b_{1}) d x = (c_{2} - c_{1}) (b_{2} - b_{1}) \int_{a_{1}}^{a_{2}} d x = (c_{2} - c_{1}) (b_{2} - b_{1}) (a_{2} - a_{1}) = (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) = R H S . H e n c e, p r o v e d .$

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LetR=x,y,z|a1⩽x⩽a2,b1⩽y⩽b2,c1⩽z⩽c2.Provethat:∭RdV=a2-a1b2-b1c2-c1.WhatistherelationshipbetweenRandtheproducta2-a1b2-b1c2-c1.

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Step 1. Given Information.

Step 2. Proof.

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