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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Use cylindrical coordinates.

25. (a) Find the volume of the region E that lies between the paraboloid z = 24 - x2 - y2 and the cone Z = 2 x 2 + y 2 .

(b) Find the centroid of E (the center of mass in the case where the density is constant).

(a)

To determine

To find: The volume of the given solid by using cylindrical coordinates.

Explanation

Given:

The region E lies between the paraboloid z=24x2y2 and the cone z=2x2+y2 .

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 equations, the value of z is obtained below.

z=24x2y2z=24r22r=24r2r2+2r24=0

The above equation is a quadratic equation.

Thus, solve the equation and obtain the value of r=4 .

Therefore, r varies from 0 to 4, and θ varies from 0 to 2π and z varies from 2r to z=24r2 .

Use the formula mentioned above to change the given problem into cylindrical coordinates. Then, by the equation (1), the volume of the given solid is computed as follows.

Ef(x,y,z)dV=0402π2r24r2(r)dzdθdr=0402π2r24r2rdzdθdr

Integrate with respect to z and apply the limit of it

(b)

To determine

To find: The centroid of the given region E.

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