   Chapter 15, Problem 33RE

Chapter
Section
Textbook Problem

Calculate the value of the multiple integral.33. ∭ E y z   d V , where E lies above the plane z = 0, below the plane z = y, and inside the cylinder x2 + y2 = 4

To determine

To calculate: The given triple integral.

Explanation

Given:

The function is f(x,y,z)=yz .

The region E lies above the plane z=0 and below the plane z=y and inside the cylinder x2+y2=4 .

Formula used:

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)

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

abcdefg(x)h(y)k(z)dzdydx=abg(x)dxcdh(y)dyefk(z)dz (2)

Calculation:

Since the equation of the cylinder is given, solving with cylindrical coordinates is a wise choice. From the given conditions, r varies from 0 to 2, θ varies from 0 to π and z varies from 0 to y and y=rsinθ in cylindrical coordinates. Therefore, z varies from 0 to rsinθ . Thus, by the equation (1), the value of the integral becomes,

EyzdV=0π020rsinθrsinθz(r)dzdrdθ=0π020rsinθr2sinθzdzdrdθ

First compute the integral with respect to z and apply the limit.

EyzdV=0π02r2sinθ(z22)0rsinθdrdθ=0π02r2sinθ((rsinθ)22(0)22)drdθ=120π02r2sinθ(r2sin2θ)drdθ=120π02r4sin3θdrdθ

Use the equation (2) to separate the integral

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