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Chapter 14, Problem 12RE
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### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Changing the Order of Integration In Exercises 11-14, sketch the region K whose area is given by the iterated integral. Then change the order of integration and show that both orders yield the same area. ∫ − 3 3 ∫ 0 9 − y 2 d x   d y

To determine

To calculate: The region R, whose area is given by the following iterated integral 3309y2dxdy. Also,

change the order of integration and show that bothorders yield the same area.

Explanation

Graph:

Inner limit is given as 0â‰¤xâ‰¤9âˆ’y2 it means Region R is bounded on the left by the y axis and on the right by parabola x=9âˆ’y2.

Outer limit is given as âˆ’3â‰¤yâ‰¤3 it means Region R is bounded below by the line x=âˆ’3

and above by the line x=3.

To plot x=9âˆ’y2 convert it into its vertex form i.e. (xâˆ’9)=âˆ’(yâˆ’0)2.

Its vertex V is (9,0).

Area is given by the shaded region of the graph

Calculation:

Find area without changing the order of the integral:

âˆ«âˆ’33âˆ«09âˆ’y2dxdy=âˆ«âˆ’33[x]09âˆ’y2dy=âˆ«âˆ’33(9âˆ’y2)dy=[9yâˆ’y33]âˆ’33=((9â‹…3)âˆ’333)âˆ’((9â‹…(âˆ’3))+333)

Further solved ahead as:

âˆ«âˆ’33âˆ«09âˆ’y2dxdy=(27x

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