Chapter 13: Q 73. (page 1006)

Let $a < b and c < d$ be real numbers and $R$ be rectangle defined by
$R = {(x, y) ∣ a \leq x \leq b and c \leq y \leq d}$ in $x y$ plane. If $g (x)$ is continuous in interval $[a, b]$ and $h (y)$ is continuous on $[c, d]$
Use Fubini's theorem to prove that $\iint_{R} g (x) h (y) d A = (\int_{a}^{b} g (x) d x) (\int_{c}^{d} h (y) d y) .$

Short Answer

Expert verified

It can be proved using definition of Fubini's theorem

$\iint_{R} g (x) h (y) d A = \int_{a}^{b} \int_{c}^{d} g (x) h (y) d y d x$ .

Step by step solution

Given Information

It is given that $a < b and c < d$ , $a, b, c, d$ are real numbers.

Rectangle is defined by

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

Fubini's Theorem

It states that $\iint_{R} g (x) h (y) d A = \int_{a}^{b} \int_{c}^{d} g (x) h (y) d y d x$

Treating $x$ as constant

$\iint_{R} g (x) h (y) d A = \int_{a}^{b} \int_{c}^{d} g (x) h (y) d y d x$

$= \int_{a}^{b} (\int_{c}^{d} g (x) h (y) d y) d x$

$= \int_{a}^{b} g (x) (\int_{c}^{d} h (y) d y) d x$

$= \int_{a}^{b} g (x) d x (\int_{c}^{d} h (y) d y)$

$\Rightarrow \iint_{R} g (x) h (y) d A = (\int_{a}^{b} g (x) d x) (\int_{c}^{d} h (y) d y)$

Hence proved

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Fubini's Theorem

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Let a<bandc<dbe real numbers and Rbe rectangle defined byR={(x,y)∣a≤x≤bandc≤y≤d}in xyplane. If g(x)is continuous in interval [a,b]and hyis continuous on [c,d]Use Fubini's theorem to prove that∬Rg(x)h(y)dA=∫abg(x)dx∫cdh(y)dy.

Short Answer

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Given Information

Fubini's Theorem

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