   Chapter 14, Problem 12RE

Chapter
Section
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 0x9y2 it means Region R is bounded on the left by the y axis and on the right by parabola x=9y2.

Outer limit is given as 3y3 it means Region R is bounded below by the line x=3

and above by the line x=3.

To plot x=9y2 convert it into its vertex form i.e. (x9)=(y0)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:

3309y2dxdy=33[x]09y2dy=33(9y2)dy=[9yy33]33=((93)333)((9(3))+333)

Further solved ahead as:

3309y2dxdy=(27x

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