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Chapter 13: Q 73. (page 1006)

Let a<bandc<dbe real numbers and Rbe rectangle defined by

R={(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

Expert verified

It can be proved using definition of Fubini's theorem

∬Rg(x)h(y)dA=∫ab∫cdg(x)h(y)dydx.

Step by step solution

01

Given Information

It is given that a<bandc<d, a,b,c,dare real numbers.

Rectangle is defined by

R={(x,y)∣a≤x≤bandc≤y≤d}

02

Fubini's Theorem

It states that ∬Rg(x)h(y)dA=∫ab∫cdg(x)h(y)dydx

Treating xas constant

∬Rg(x)h(y)dA=∫ab∫cdg(x)h(y)dydx

=∫ab∫cdg(x)h(y)dydx

=∫abg(x)∫cdh(y)dydx

=∫abg(x)dx∫cdh(y)dy

⇒∬Rg(x)h(y)dA=∫abg(x)dx∫cdh(y)dy

Hence proved

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