Chapter 13: Q 65. (page 1014)

Let $f (x, y)$ be an integrable function on the rectangle $R = {(x, y) ∣ a \leq x \leq b and c \leq y \leq d}$ and $c \leq y \leq d}$ , and let $α \in ℝ$ . Use the definition of the double integral to prove that
$\iint_{R} α f (x, y) d A = α \iint_{R} f (x, y) d A .$

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 $\iint_{R} α f (x, y) d A = α \iint_{R} f (x, y) d A$ and $R = {(x, y) ∣ a \leq x \leq b and c \leq y \leq d}$

$α$ is real number.

Proof

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

and $Δ = \sqrt{(Δ x)^{2} + (Δ y)^{2}}$

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

$= α \iint_{R} f (x, y) d A$

The equation is true.

Changing order of sum

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

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

$= α \iint_{R} f (x, y) d A$

Equation is true.

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Let $f (x, y)$ be an integrable function on the rectangle $R = {(x, y) ∣ a \leq x \leq b and c \leq y \leq d}$ and $c \leq y \leq d}$ , and let $α \in ℝ$ . Use the definition of the double integral to prove that
$\iint_{R} α f (x, y) d A = α \iint_{R} f (x, y) d A .$

Short Answer

Step by step solution

Given Information

Proof

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Let f(x,y)be an integrable function on the rectangle R={(x,y)∣a≤x≤bandc≤y≤d}and c≤y≤d}, and let α∈ℝ. Use the definition of the double integral to prove that∬Rαf(x,y)dA=α∬Rf(x,y)dA.

Short Answer

Step by step solution

Given Information

Proof

One App. One Place for Learning.

Most popular questions from this chapter

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Let $f (x, y)$ be an integrable function on the rectangle $R = {(x, y) ∣ a \leq x \leq b and c \leq y \leq d}$ and $c \leq y \leq d}$ , and let $α \in ℝ$ . Use the definition of the double integral to prove that
$\iint_{R} α f (x, y) d A = α \iint_{R} f (x, y) d A .$