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I have difficulties in integrating this scalar function over the assigned volume.

Let $D=\{(x,y,z):(x-2z)^2+(y-x)^2+(x+z)^2\le4,\,0\le x+y+z\le1\}$

Calculate $\int_D z\,dxdydz$

Fuzzee Lowgeek
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1 Answers1

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Hint: let $u=x-2z$, $v=y-x $, $w=x+z$. Solving for $ x, y, z $, so that your new domain becomes : $$ D_1=\{(u, v, w)| 0 \le u^2+v^2+w^2\le 4\}$$ with $$0\le u +3v+5w\le 3$$

Finally, $$I=\int\int\int_{D_1}|J (u, v, w)|\cdot \frac {(w-u)}{3} dudvdw$$

Where $|J(.,.,.)|$ is the Jacobian. I think you can continue. . .

Mohammad W. Alomari
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