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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 the transformation x = u2, y = v2 z = w2 to find the volume of the region bounded by the surface x + y + z = 1 and the coordinate planes.

To determine

To find: The volume of the region bounded by the surface x+y+z=1 and the coordinate planes by using the transformation x=u2 , y=v2 and z=w2 .

Explanation

Formula used:

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

The volume of the region is V=EdV .

Calculation:

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

If x=u2 , then xu=2u , xv=0 and xw=0 .

If y=v2 , then yu=0 , yv=2v and yw=0 .

If z=w2 , then zu=0 , zv=0 and zw=2w .

Compute the Jacobian value.

(x,y,z)(u,v,w)=|2u0002v0002w|=2u((2v)(2w)0)0(00)+0(00)=2u(4vw)+0+0=8uvw

Rewrite the given transformations as, u=x , v=y and w=z . So, the region bounded by the surface is u+v+w=1

Set w=0 .

Therefore, v varies from 0 to 1u , w varies from 0 to 1uv and u varies from 0 to 1.

Then, the given integral is, V=0101u01uv8uvwdwdvdu .

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

V=80101uuv[w22]01uvdvdu=80101uuv[(1uv)22(0)2]dvdu=40101uuv(1uv)2dvdu

Simplify the above integral.

V=40101uuv[(1u)v]2dvdu=40101uuv[(1u)2+v22(1u)v]dvdu=40101u[u(1u)2v+uv32u(1u)v2]dvdu

Integrate the above integral with respect to v and apply the limit

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Chapter 15 Solutions

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