   Chapter 15, Problem 38RE

Chapter
Section
Textbook Problem

Find the volume of the given solid.38. Bounded by the cylinder x2 + y2 = 4 and the planes z = 0 and y + z = 3

To determine

To find: The volume of the given solid.

Explanation

Given:

The region E is bounded by the cylinder x2+y2=4 and the planes z=0,y+z=3 .

Formula used:

The volume of the given solid is given by, V=EdV .

If f is a cylindrical region E given by h1(θ)rh2(θ),αθβ, u1(rcosθ,rsinθ)zu1(rcosθ,rsinθ) where 0βα2π , then,

Ef(x,y,z)dV=αβh1(θ)h2(θ)u1(rcosθ,rsinθ)u2(rcosθ,rsinθ)f(rcosθ,rsinθ,z)rdzdrdθ (1)

Calculation:

Since equation of the cylinder is given, it is much easier to use cylindrical coordinates. From the given conditions, it is observed that r varies from 0 to 2, z varies from 0 to 3y and θ varies from 0 to 2π . In cylindrical coordinates, y=rsinθ . Then, the volume of the given solid by the equation (1) is,

V=EdV=02π0203rsinθdzdrdθ

Integrate with respect to z and apply the limit

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