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I am given with the following question: A) $V_1 = \{ x^2 + y^2 \leq 4 , 0\leq z\leq 3 \sqrt{x^2 + y^2 } , x\geq 0 \} $ , and I need to represent the triple integral $\int \int \int_{V_1} f(z) dxdydz$ as $\int _{?} ^{?} ?f(z) dz $ (where I need to fill in the question marks).

B) DO the same thing with the triple integral of $xf(z)$ over the region $V= \{ x^2 + (y-2)^2 \leq 4 , 0\leq z \leq 3\ \sqrt{x^2+(y-2)^2 } \} $

As for part A: I know how to write the region in cylindrical coordinates, and I know that in the $y-z$ plane : $0\leq z \leq 3y$ , in the $x-y $ plane we have $ x\leq \sqrt{4-y^2} $ and in the $x-z$ pland: $ 0\leq z \leq 3x$ . But how does this help me ? Will you please explain to me how can I solve this question?

Thanks !

homogenity
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1 Answers1

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$$\iint_{z^2/9\leqslant x^2+y^2\leqslant4}\mathrm dx\mathrm dy=\pi\,\left(4-\tfrac19z^2\right)^+$$ $$\text{hence}$$ $$\int\!\!\!\!\int\!\!\!\!\int_{V_1}f(z)\mathrm dx\mathrm dy\mathrm dz=\pi\int_0^6\left(4-\tfrac19z^2\right)f(z)\mathrm dz$$

Did
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