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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Changing the Order of Integration In Exercises 37-44, sketch the region R whose area is given by the double integral. Then change the order of integration and show that both orders yield the same value. See Example 5.

0 1 y 2 y 3 d x   d y

To determine

To graph: The region whose area is given by double integration 01y2y3dxdy, the change the order of integration and shows both orders yield same value.

Explanation

Given Information:

The provided double integration is 01y2y3dxdy.

Graph:

Consider the double integration,

01y2y3dxdy.

From limits of integration, the bounds for x are y2xy3 and bounds for y are 0y1.

The table shown the coordinate of x=y3 ,

x -Coordinates y - Coordinates (x,y) Coordinates
0 2 (0,2)
0 2 (0,2)
4 0 (4,0)

The table shown the coordinate of y2=x,

x -Coordinates y - Coordinates (x,y) Coordinates
0 0 (0,0)
1 1 (1,1)

The graph of region bounded by y2xy3 and 0y1 is shown in below,

The area for the region y2xy3 and 0y1 is

01y2y3dxdy

Evaluate the above integration integrate with respect to x by holding y constant,

01y2y3dxdy=01[x]y2y3dy

Now, replace the x by limit of integration,

01[x]y2y3dy=01[y3y2]dy

Evaluate the above integration integrate with respect to y by holding x constant,

01[y3y2

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