   Chapter 14.1, Problem 47E

Chapter
Section
Textbook Problem

Changing the Order of Integration In Exercises 45-50. sketch the region R of integration and change the order of integration. ∫ 0 10 ∫ 0 ln y f ( x , y )   d x   d y

To determine

To graph: The region R for the given iterated integral 1100lnyf(x,y)dxdy. Further, change the order of integration.

Explanation

Given:

The iterated integral, 1100lnyf(x,y)dxdy.

Graph:

Take the provided iterated integral into consideration.

1100lnyf(x,y)dxdy

In the provided integral, limit of x is 0xlny and limit of y is 1y10.

To plot the equation x=lny, i.e., the equation of a logarithmic curve, make the following table:

 x 1 10 5 y 0 ln10 ln5

Step 1: Draw the graph of inequality xlny and x0 as follows:

Step 2: Draw the graph of inequality y1 and y10 as follows:

Step 3: Merge the two graphs above to sketch the area calculated by the expression 1100lnyf(x,y)dxdy.

Interpretation:

From the graph, it can be observed that there are two limits present, i.e., the outside and inside limits.

In this case, the outer limit shows that value of y ranges from 1 to 10 while the inner limit x varies from 0 to lny

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