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Chapter 13: Q 65. (page 1014)

Let f(x,y)be an integrable function on the rectangle R={(x,y)鈭�a鈮�x鈮�bandc鈮�y鈮�d}and c鈮�y鈮�d}, and let 伪鈭�鈩�. Use the definition of the double integral to prove that

鈭�R伪f(x,y)dA=伪鈭�Rf(x,y)dA.

Short Answer

Expert verified

To prove this, write the double integral on left hand side as double Reimann sum.

Step by step solution

01

Given Information

It is given that 鈭�R伪f(x,y)dA=伪鈭�Rf(x,y)dAand R={(x,y)鈭�a鈮�x鈮�bandc鈮�y鈮�d}

伪is real number.

02

Proof

鈭�R伪f(x,y)dA=lim螖鈫�0鈭�i=1m鈭�j=1n伪fxi*,yj*螖A

and 螖=(螖x)2+(螖y)2

鈭�R伪f(x,y)dA=伪lim螖鈫�0鈭�i=1m鈭�j=1nfxi*,yj*螖A

=伪鈭�Rf(x,y)dA

The equation is true.

Changing order of sum

鈭�R伪f(x,y)dA=lim螖鈫�0鈭�j=1n鈭�i=1m伪fxi*,yj*螖A

鈭�R伪f(x,y)dA=伪lim螖鈫�0鈭�j=1n鈭�i=1mfxi*,yj*螖A

=伪鈭�Rf(x,y)dA

Equation is true.

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Most popular questions from this chapter

Identify the quantities determined by the integral expressions in Exercises 19鈥�24. If x, y, and z are all measured in centimeters and 蚁(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.

鈭�惟鈥�x蚁(x,y,z)dV

What is the difference between a double integral and an iterated integral?

Evaluate the iterated integral :

鈭�01鈥�鈭�24鈥�鈭�鈭�15鈥�x+yz2dxdydz

Identify the quantities determined by the integral expressions in Exercises 19鈥�24. If x, y, and z are all measured in centimeters and 蚁(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.

鈭�惟鈥�x2+z2蚁(x,y,z)dV

Expressthesum3e4+3e9+3e16+4e4+4e9+4e16usingdouble-summationnotation.
See all solutions

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