Chapter 13: Q. 73 (page 1057)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information.

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

Step 2. Proof.

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

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Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof.

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

Decision Maths

Pure Maths

Statistics

Theoretical and Mathematical Physics

Geometry

Calculus

Study anywhere. Anytime. Across all devices.