   Chapter 15.3, Problem 4E

Chapter
Section
Textbook Problem

A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write ∬ R f ( x ,   y )   d A as an iterated integral, where f is an arbitrary continuous function on R.4. To determine

To write: An iterated integral by deciding whether to use polar coordinates or rectangular coordinates.

Explanation

Formula used:

If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π , then, Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

Calculation:

Since the given region is the semicircle, it is easier to use polar coordinates.

From the figure given in the problem, it is observed that radius r ranges from 0 to 3 and θ varies from π4 to 3π4 .

Therefore, the polar coordinates is, R={(r,θ)|0r3,π4θ3π4}

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