   Chapter 7.8, Problem 41E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### 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 2 ∫ x / 2 1 d y   d x

To determine

To graph: The region whose area is given by double integration 02x/21dydx, the change the order of integration and shows both orders yield same value.

Explanation

Given Information:

The provided double integration is 02x/21dydx.

Graph:

Consider the double integration,

02x/21dydx.

From limits of integration, the bounds for x are 0x2 and bounds for y are x2y1.

The table shown the coordinate of y=x2,

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

The graph of region bounded by 0x2 and x2y1 is shown in below,

The area for the region 0x2 and x2y1 is

02x/21dydx

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

02x/21dydx=02[y]x/21dx

Now, replace the y by limit of integration,

02[y]x/21dx=02[1x2]dx

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

02

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