   Chapter 15.3, Problem 21E

Chapter
Section
Textbook Problem

Use polar coordinates to find the volume of the given solid.21. Below the plane 2x + y + z = 4 and above the disk x2 + y2 ≤ 1

To determine

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

Explanation

Given:

The region D lies below the plane 2x+y+z=4 and above the circle x2+y21 .

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:

From the given region D, it is observed that the value of r varies from 0 to 1 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.

DzdA=02π01(42rcosθrsinθ)rdrdθ=02π01(4r2r2cosθr2sinθ)drdθ

Integrate the function with respect to r and apply the limit.

02π01(4r2r2cosθr2sinθ)drdθ=02π[4r222r3cosθ3r3sinθ3]01dθ=

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