Chapter 13: Q 67. (page 1016)

Prove Theorem 13.10 (a). That is, show that if $f (x, y)$ is an integrable function on the general region and $c \in R$ , then
$\iint_{Ω} α f (x, y) d A = α \iint_{Ω} f (x, y) d A$

Short Answer

Expert verified

To prove this, write the double integral on left hand side as double Reimann sum.

Step by step solution

Given Information

It is given that

$\iint_{Ω} c f (x, y) d A = c \iint_{Ω} f (x, y) d A$

Region is subset of rectangular region defined by

role="math" localid="1653943372914" $R = {(x, y) ∣ a \leq x \leq b and c \leq y \leq d}, that is, Ω \subseteq R and c$ is real number.

Simplify using property

We know property of double integral

$\iint_{Ω} f (x, y) d A = \iint_{R} F (x, y) d A$

and

localid="1653944299120" $F (x, y) = (x, y), if (x, y) \in Ω$ and $0, if (x, y) \notin Ω$

write the double integral on left hand side as double Reimann sum.

$\iint_{R} c F (x, y) d A = \lim_{Δ \to 0} \sum_{i = 1}^{m} \sum_{j = 1}^{n} c F (x_{i}^{*}, y_{j}^{*}) Δ A$ and

$Δ = \sqrt{(Δ x)^{2} + (Δ y)^{2}}$

Simplify RHS

$\iint_{R} c F (x, y) d A = c \lim_{Δ \to 0} \sum_{i = 1}^{m} \sum_{j = 1}^{n} F (x_{i}^{*}, y_{j}^{*}) Δ A$

$= c \iint_{R} F (x, y) d A$

From same property $\iint_{Ω} c f (x, y) d A = c \iint_{Ω} f (x, y) d A$

Equation is true.

Simplification

Changing order of sum

$\iint_{R} c F (x, y) d A = \lim_{Δ \to 0} \sum_{j = 1}^{n} \sum_{i = 1}^{m} c F (x_{i}^{*}, y_{j}^{*}) Δ A$

$\iint_{R} c F (x, y) d A = c \lim_{Δ \to 0} \sum_{j = 1}^{n} \sum_{i = 1}^{m} F (x_{i}^{*}, y_{j}^{*}) Δ A$

$= c \iint_{R} F (x, y) d A$

From property stated above

$\iint_{Ω} c f (x, y) d A = c \iint_{Ω} f (x, y) d A$

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Prove Theorem 13.10 (a). That is, show that if $f (x, y)$ is an integrable function on the general region and $c \in R$ , then
$\iint_{Ω} α f (x, y) d A = α \iint_{Ω} f (x, y) d A$

Short Answer

Step by step solution

Given Information

Simplify using property

Simplification

One App. One Place for Learning.

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Prove Theorem 13.10 (a). That is, show that if f(x,y)is an integrable function on the general region and c∈R, then∬Ωαf(x,y)dA=α∬Ωf(x,y)dA

Short Answer

Step by step solution

Given Information

Simplify using property

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Logic and Functions

Applied Mathematics

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

Prove Theorem 13.10 (a). That is, show that if $f (x, y)$ is an integrable function on the general region and $c \in R$ , then
$\iint_{Ω} α f (x, y) d A = α \iint_{Ω} f (x, y) d A$