Q. 72 Let f(x)be an integrable functio... [FREE SOLUTION]

Chapter 13: Q. 72 (page 1057)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information.

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

Step 2. Proof.

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

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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)be an integrable function on the rectangular solid R=x,y,z|a1猢�x猢�a2,b1猢�y猢�b2,c1猢�z猢�c2, and let 魏鈭�鈩�.Use the definition of the triple integral to prove that:鈭�R魏f(x,y,z)dV=魏鈭�Rf(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

Calculus

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Study anywhere. Anytime. Across all devices.