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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Set up, but do not evaluate, integral expressions for (a) the mass, (b) the center of mass, and (c) the moment of inertia about the z-axis.

48. The hemisphere x2 + y2 + z2 ≤ 1, z ≥ 0; ρ (x, y, z) = x 2 + y 2 + z 2

(a)

To determine

To find: The integral expression for the mass and not to evaluate the value.

Explanation

Given:

The hemisphere is, x2+y2+z21 , z0 and the density is given by ρ(x,y,z)=x2+y2+z2 .

Calculation:

Mass of the solid, mass(m)=density×volume . Here the density is given by ρ(x,y,z)=x2+y2+z2 .

The mass of the solid is, m=liml,m,ni=1lj=1mk=1nρ(xij*,yij*,zij*)ΔA=Eρ(x,y,z)dA .

Here, the density function is given by ρ(x,y,z) and E is the region that is occupied by the solid.

From the given, x2+y2+z21 then the value of z, z=±1x2y2 but take the upper hemisphere z=1x2y2 and z0

(b)

To determine

To find: The integral expression for the center of mass and not to evaluate the value.

(c)

To determine

To find: The integral expression for the moment of inertia about the z-axis and not to evaluate the value.

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