   Chapter 15, Problem 13P

Chapter
Section
Textbook Problem

The plane x a + y b + z c = 1   a > 0 ,   b > 0 ,   c > 0 cuts the solid ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 ≤ 1 into two pieces. Find the volume of the smaller piece.

To determine

To find: The volume of smaller piece when the plane cuts the ellipsoid.

Explanation

Given:

The equation of the plane is xa+yb+zc=1 .

The condition satisfied by the plane is a>0,b>0 and c>0 .

The equation of the ellipsoid is x2a2+y2b2+z2c21 .

Formula used:

The volume of the given region is V=RdV=S|(x,y,z)(u,v,w)|dV (1)

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

Calculation:

Let the transformations be x=au , y=bv and z=cw .

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.

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

Substitute the transformations in the ellipsoid of the region E,

(au)2a2+(bv)2b2+(cw)2c21a2u2a2+b2v2b2+c2w2c21u2+v2+w21

The equation u2+v2+w21 to the ellipsoid x2a2+y2b2+z2c21 , which is in the form of the sphere centered at origin with the radius 1 and also maps u+v+w=1 to xa+yb+zc=1 .

To compute the volume of the sphere, consider the plane cuts the upper part of the sphere with the radius r and the smaller sliced part produces height h.

By using the spherical coordinates, the region of smaller slice of the sphere is obtained as,

{(ρ,θ,ϕ)|(rh)cosϕρr,0θ2π,0ϕcos1(rh)r}

Then, the volume of the integral is, V=02π0cos1((rh)r)(rh)cosϕrρ2sinϕdρdϕdθ .

Integrate the above integral with respect to ρ and apply the limit.

V=02π0cos1((rh)r)[ρ33](rh)cosϕrsinϕdϕdθ=02π0cos1((rh)r)((r)33((rh)cosϕ)33)sinϕdϕdθ=1302π0cos1((rh)r)(r3(rh)3cos3ϕ)sinϕdϕdθ

Let u=cosϕ

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