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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∬Rf(x,y)dA=∫cd∫abf(x,y)dxdy=∫ab∫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)∣a≤x≤band c≤y≤d}and let f(x,y)be continuous on R. Then Fubini's theorem states as

∬Rf(x,y)dA=∫cd∫abf(x,y)dxdy=∫ab∫cdf(x,y)dydx

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Most popular questions from this chapter

Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.

∫-15∫-32∫48f(x,y,z)dzdxdy

Evaluate the iterated integral :

∫0π/2 ∫0cos⁡y ∫0sin⁡y (2x+y)dzdxdy

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

In Exercises 61–64, let R be the rectangular solid defined by

R=(x,y,z)|0≤a1≤x≤a2,0≤b1≤y≤b2,0≤c2≤z≤c2.

Assume that the density of R is uniform throughout.

(a) Without using calculus, explain why the center of

mass is a1+a22,b1+b22,c1+c22.

(b) Verify that a1+a22,b1+b22,c1+c22is the center of mass by using the appropriate integral expressions.

Evaluate Each of the integrals in exercises 33-36 as an iterated integral and then compare your answer with thoise you found in exercise 29-32

∫∫RxydAWhereR={(x,y)|0≤x≤2,1≤y≤4}
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