   Chapter 15.7, Problem 29E

Chapter
Section
Textbook Problem

Evaluate the integral by changing to cylindrical coordinates.29. ∫ − 2 2 ∫ − 4 − y 2 4 − y 2 ∫ x 2 + y 2 2 x z   d z   d x   d y

To determine

To evaluate: The integral by changing to cylindrical coordinates.

Explanation

Given:

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

The rectangular coordinates of the given triple integral are {(x,y,z)|2y2,4y2x4y2,x2+y2z2} .

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)

The cylindrical coordinates (r,θ,z) corresponding to the rectangular coordinates (x,y,z) is,

r=x2+y2θ=tan1(yx)z=z

Calculation:

Substitute x=rcosθ,y=rsinθ,z=z in the given function f(x,y,z) .

f(x,y,z)=xzf(r,θ,z)=zrcosθ

The limits become,

x=4y2x2+y2=4r2=4r=±2

And

z=x2+y2z=r2z=r

From the limits of x given, it is observed that it is a circle of radius 2

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