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In Exercises 29–34, sketch the region determined by the limits of the iterated integrals and then give another iterated integral (or a sum of iterated integrals if necessary) using the opposite order of integration.

12lnxexf(x,y)dydx

Short Answer

Expert verified

The sketch of the region is;

The iterated integral:

011eyf(x,y)dxdy+1e12f(x,y)dxdy+ee2lny2f(x,y)dxdy

Step by step solution

01

Step 1. Given information

Integral:

12lnxexf(x,y)dydx

02

Step 2. Sketch the region:

In the given integral we can see that the region is :

lnxyex1x2

So the sketch is :

03

Step 3. Write another iterated integral.

When y=lnxthen x=ey

When y=exthen x=lny

The region is divided into three parts:

When x=1then y=ln1=0

When x=1then y=e1=e

Whenx=2thenlocalid="1652103242499" y=ln2,e2

The integral can be changed as:

localid="1652159180867" 011eyf(x,y)dxdy+1e12f(x,y)dxdy+ee2lny2f(x,y)dxdy

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