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

R={(x,y)axbandcyd}in xyplane. If g(x)is continuous in interval [a,b]and hyis continuous on [c,d]

Use Fubini's theorem to prove thatRg(x)h(y)dA=abg(x)dxcdh(y)dy.

Short Answer

Expert verified

It can be proved using definition of Fubini's theorem

Rg(x)h(y)dA=abcdg(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)axbandcyd}

02

Fubini's Theorem

It states that Rg(x)h(y)dA=abcdg(x)h(y)dydx

Treating xas constant

Rg(x)h(y)dA=abcdg(x)h(y)dydx

=abcdg(x)h(y)dydx

=abg(x)cdh(y)dydx

=abg(x)dxcdh(y)dy

Rg(x)h(y)dA=abg(x)dxcdh(y)dy

Hence proved

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