   Chapter 15.3, Problem 20E

Chapter
Section
Textbook Problem

Use polar coordinates to find the volume of the given solid.20. Below the cone z = x 2 + y 2 and above the ring 1 ≤ x2 + y2 ≤ 4

To determine

To find: The volume of the given solid by using polar coordinates.

Explanation

Given:

The region D lies below the cone z=x2+y2 and above the annulus 1x2+y24 .

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)

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (2)

Calculation:

From the given region D, it is observed the value of r varies from 1 to 2 and the value of θ varies from 0 to 2π .

Substitute x=rcosθ and y=rsinθ in the equation (1) and obtain the required volume

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