   Chapter 15.9, Problem 21E

Chapter
Section
Textbook Problem

(a) Evaluate ∫∫∫E dV, where E is the solid enclosed by the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1. Use the transformation x = au, y = bv, z = cw.(b) The earth is not a perfect sphere; rotation has resulted in flattening at the poles. So the shape can be approximated by an ellipsoid with a = b = 6378 km and c = 6356 km. Use part (a) to estimate the volume of the earth.(c) If the solid of part (a) has constant density k. find its moment of inertia about the z-axis.

(a)

To determine

To evaluate: The integral EdV .

Explanation

Given:

The region E is enclose by the ellipsoid x2a2+y2b2+z2c2=1 with the transformation x=au , y=bv and z=cw .

Formula used:

The Jacobian value J is, (x,y,z)(u,v,w)=|xuxvxwyuyvywzuzvzw|

Calculation:

Find the partial derivative of x, y and z with respect to u, v and w, respectively.

If x=au , then xu=a , xv=0 and xw=0 .

If y=bv , then yu=0 , yv=b and yw=0 .

If z=cw , then zu=0 , zv=0 and zw=c .

Obtain the Jacobian value J.

(x,y,z)(u,v,w)=|a000b000c|=a(bc0)0(00)+0(00)=a

(b)

To determine

To estimate: The volume of the earth.

(c)

To determine

To find: The moment of inertia of the solid from part (a) is, EdV about the z-axis with constant density k.

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