   Chapter 15.7, Problem 17E

Chapter
Section
Textbook Problem

Use cylindrical coordinates.17. Evaluate ∭ E x 2 + y 2   d V , where E is the region that lies inside the cylinder x2 + y2 = 16 and between the planes z = –5 and z = 4.

To determine

To evaluate: The given triple integral by using cylindrical coordinates.

Explanation

Given:

The function is f(x,y,z)=x2+y2 .

The region E lies inside the cylinder x2+y2=16 and the planes z=5,z=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)

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

r=x2+y2θ=tan1(yx)z=z

Calculation:

By the given conditions, it is observed that r varies from 0 to 4, θ varies from 0 to 2π and z varies from 5 to 4

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