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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.

2 2 0 4 y 2 d x   d y

To determine

To graph: The region whose area is given by double integration 2204y2dxdy, the change the order of integration and shows both orders yield same value.

Explanation

Given Information:

The provided double integration is 2204y2dxdy.

Graph:

Consider the double integration,

2204y2dxdy.

From limits of integration, the bounds for x are 0x4y2 and bounds for y are 2y2.

The table shown the coordinate of x=4y2,

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

The graph of region bounded by 0x4y2and 2y2 is shown in below,

The area for the region 0x4y2and 2y2 is

2204y2dxdy

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

2204y2dxdy=22[x]04y2dy

Now, replace the x by limit of integration,

22[x]04y2dy=22[4y2]dy

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

22[4y2]dy=[4yy33]22

Now, replace the y by limit of integration,

[4yy33]22=[4(2)(2)33][4(2)(2)33]=4(2)(2)33+4(2)(2)33=16163=323

Now change the order of integration dxdy to dydx

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