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Chapter 13: Q.29 (page 991)

In Exercises 29鈥�38, find an iterated integral in polar coordinates that represents the area of the given region in the polar plane and then evaluate the integral.

29.The region enclosed by the spiral r=胃and the x-axis on the interval 0鈮�胃鈮�蟺.

Short Answer

Expert verified

Area of the region bounded by the spiral and the x- axis isA=蟺36

Step by step solution

01

Given information

The region enclosed by the spiral r=胃and the x-axis on the interval 0鈮�胃鈮�蟺.

02

Calculating the region enclosed by the spiral 

The objective of this problem is to find an iterated integral in polar coordinates that represents the area of the given region in the polar plane and then evaluate the integral.

The region is enclosed by the spiral r=胃and the x-axis on the interval 0鈮�胃鈮�蟺.

Area of the region bounded by the spiral and the x- axis can be expressed as

A=鈭�0n鈭�0r-胃rdrd胃

Integrate with respect to rfirst.

A=鈭�0*r220胃d胃A=鈭�0蟺胃2-02d胃A=胃360蟺

Area of the region bounded by the spiral and the x-axis is

A=蟺36

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