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Chapter 13: Q. 13 (page 1003)

State Fubini's theorem.

Short Answer

Expert verified

Ans: Then Fubini's theorem states as鈭�Rf(x,y)dA=鈭�cd鈭�abf(x,y)dxdy=鈭�ab鈭�cdf(x,y)dydx

Step by step solution

01

Step 1. Given information: 

Fubini's theorem.

02

Step 2. Stating Fubini's theorem:

Fubini's theorem states that a double integral can be computed using iterated integrals, which means the order of integration can be switched for which the order chosen is easier to evaluate. Let a<band c<dbe real numbers, let R be the rectangle defined by R={(x,y)鈭�a鈮�x鈮�band c鈮�y鈮�d}and let f(x,y)be continuous on R. Then Fubini's theorem states as

鈭�Rf(x,y)dA=鈭�cd鈭�abf(x,y)dxdy=鈭�ab鈭�cdf(x,y)dydx

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Most popular questions from this chapter

Let 惟be a lamina in the xy-plane. Suppose 惟is composed of n non-overlapping lamin忙 role="math" localid="1650321722341" 惟1,惟2,鈥�,惟n.Show that if the masses of these lamin忙 are m1,m2,鈥�,mnand the centers of masses are x炉1,y炉1,x炉2,y炉2,鈥�,x炉n,y炉n,then the center of mass of 惟is x炉,y炉,where

x炉=鈭�k=1nmkx炉k鈭�k=1nmkandy炉=鈭�k=1nmky炉k鈭�k=1nmk.

Explain how to construct a Riemann sum for a function of two variables over a rectangular region.

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

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

Let f(x)be an integrable function on the rectangular solid R=x,y,z|a1猢�x猢�a2,b1猢�y猢�b2,c1猢�z猢�c2, and let 魏鈭�鈩�.Use the definition of the triple integral to prove that:

鈭�R魏f(x,y,z)dV=魏鈭�Rf(x,y,z)dV.
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