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Chapter 13: Q 75. (page 1007)

Let a<band c<dbe real numbers, and let Rbe the

rectangle in the xy-plane defined by

R={(x,y)∣a≤x≤bandc≤y≤d}.

Prove that role="math" localid="1653851117688" ∬RdA=(b-a)(d-c), what is the relation between Rand product of(b-a)(d-c)?

Short Answer

Expert verified

This is evaluated using Fubini's theorem∬RdA=∫ab∫cddydx

Step by step solution

01

Given Information

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

R is rectangle in cartesian plane defined byR={(x,y)∣a≤x≤bandc≤y≤d}

02

Use Fubini's Theorem

By theorem ∬RdA=∫ab∫cddydx

Treat xas constant

=∫ab∫cddydx

=∫ab[y]y=cy=ddx

=(d-c)∫abdx

=(d-c)[x]x=ax=b

=(d-c)(b-a)

⇒∬RdA=(d-c)(b-a)

Hence proved.

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