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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 2 y 2 d y   d x

To determine

To graph: The region whose area is given by double integration 012y2dxdy, the change the order of integration and shows both orders yield same value.

Explanation

Given Information:

The provided double integration is 012y2dxdy.

Graph:

Consider the double integration,

012y2dxdy.

From limits of integration, the bounds for x are 0x1 and bounds for y are 2yxy.

The table shown the coordinate of y=3x,

x -Coordinates y - Coordinates (x,y) Coordinates
2 1 (2,1)
0 0 (0,0)

The graph of region bounded by 0x1 and 2yxy is shown in below,

The area for the region 0x1 and 2yxy is

012y2dxdy

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

012y2dxdy=01[x]2y2dy

Now, replace the x by limit of integration,

01[x]2y2dy=01[22y]dy

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

01[22y]dy=[2yy2]01

Now, replace the y by limit of integration,

[2yy2]01=(21)=1

Now change the order of integration dxdy to dydx

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