   Chapter 15.7, Problem 21E

Chapter
Section
Textbook Problem

Use cylindrical coordinates.21. Evaluate ∭ E x 2dV, where E is the solid that lies within the cylinder .x2 + y2 = l, above the plane z = 0, and below the cone z2 = 4x2 + 4y2.

To determine

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

Explanation

Given:

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

The region E lies within the cylinder x2+y2=1 above the plane z=0 and below the cone z2=4x2+4y2 .

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)

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

r=x2+y2θ=tan1(yx)z=z

Calculation:

From the given conditions, it is observed that r varies from 0 to 1, θ varies from 0 to 2π and z varies from 0 to 2r .

Use the formula mentioned above to change the given problem into cylindrical coordinates. Then, by the equation (1), the value of the given triple integral is obtained as follows.

Ef(x,y,z)dV=0102π02rx2(r)dzdθdr=0102π02r(rcosθ)2(r)dzdθdr=0102π02rr2cos2θ(r)dzdθdr=0102π02rr3cos2θdzdθdr

Integrate with respect to z and apply the limit of it

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