Q. 73 Let f(x, y, z) and g(x, y, z) be... [FREE SOLUTION]

Chapter 13: Q. 73 (page 1057)

Let f(x, y, z) and g(x, y, z) be integrable functions on the rectangular solid $R = \{(x, y, z) | a_{1} 猢� x 猢� a_{2}, b_{1} 猢� y 猢� b_{2}, c_{1} 猢� z 猢� c_{2}\}$ . . Use the definition of the triple integral to prove that :
${鈭�}_{R} (f (x, y, z) + g (x, y, z)) d V = {鈭�}_{R} f (x, y, z) d V + {鈭�}_{R} g (x, y, z) d V .$

Short Answer

Expert verified

${鈭�}_{R} f (x, y, z) d V = \lim_{鈭� 鈫� 0} {鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} f (x_{i}^{*}, y_{j}^{*}, z_{k}^{*}) d V . R e p l a c e f (x, y, z) + g (x, y, z) w i t h f (x, y, z) i n a b o v e, = \lim_{鈭� 鈫� 0} {鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} f (x_{i}^{*}, y_{j}^{*}, z_{k}^{*}) d V + \lim_{鈭� 鈫� 0} {鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} g (x_{i}^{*}, y_{j}^{*}, z_{k}^{*}) d V T h e r e f o r e, {鈭�}_{R} (f (x, y, z) + g (x, y, z)) d V = {鈭�}_{R} f (x, y, z) d V + {鈭�}_{R} g (x, y, z) d V . H e n c e, p r o v e d .$

Step by step solution

Step 1. Given Information.

Given: $R = \{(x, y, z) | a_{1} 猢� x 猢� a_{2}, b_{1} 猢� y 猢� b_{2}, c_{1} 猢� z 猢� c_{2}\}$

Step 2. Proof.

$A s w e k n o w f r o m t h e d e f i n i t i o n o f t r i p l e i n t e g r a l s : {鈭�}_{R} f (x, y, z) d V = \lim_{鈭� 鈫� 0} {鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} f (x_{i}^{*}, y_{j}^{*}, z_{k}^{*}) d V . R e p l a c e f (x, y, z) + g (x, y, z) w i t h f (x, y, z) i n a b o v e, {鈭�}_{R} (f (x, y, z) + g (x, y, z)) d V = \lim_{鈭� 鈫� 0} {鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} (f + g) (x_{i}^{*}, y_{j}^{*}, z_{k}^{*}) d V = \lim_{鈭� 鈫� 0} {鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} (f (x_{i}^{*}, y_{j}^{*}, z_{k}^{*}) + g (x_{i}^{*}, y_{j}^{*}, z_{k}^{*})) d V = \lim_{鈭� 鈫� 0} {鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} f (x_{i}^{*}, y_{j}^{*}, z_{k}^{*}) d V + \lim_{鈭� 鈫� 0} {鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} g (x_{i}^{*}, y_{j}^{*}, z_{k}^{*}) d V = {鈭�}_{R} f (x, y, z) d V + {鈭�}_{R} g (x, y, z) d V = R H S T h e r e f o r e, {鈭�}_{R} (f (x, y, z) + g (x, y, z)) d V = {鈭�}_{R} f (x, y, z) d V + {鈭�}_{R} g (x, y, z) d V . H e n c e, p r o v e d .$

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Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof.

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91影视

Let f(x, y, z) and g(x, y, z) be integrable functions on the rectangular solid R=x,y,z|a1猢�x猢�a2,b1猢�y猢�b2,c1猢�z猢�c2. . Use the definition of the triple integral to prove that :鈭�Rf(x,y,z)+g(x,y,z)dV=鈭�Rf(x,y,z)dV+鈭�Rg(x,y,z)dV.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Probability and Statistics

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