Finding the volume between cylinders, paraboloid and plane

Question

I need to find the volume of the solid between the cylinders $$x^{2}+y^{2}=1,\quad x^{2}+y^{2}=4$$ and limited by $$ z = 0, \quad z = 9 - x^2 + y^2 $$

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i got this far but i dont know if it is right or wrong

$$\iint\int _{0}^{9-x^2-y^2}dzdx dy$$

then turned into polar

$$\int _{0}^{2\pi}\int _1^{2}(9-r^2)r drd\theta$$

i feel like im missing something

Answer 1

You got it right. In fact, if you start directly with cylindrical coordinates, the volume is given by

$$ V = \int_1^2 \int_0^{2 \pi} \int_0^{9-r^2} r \,dz d\theta dr = 2\pi \int_1^2 r (9-r^2)\, dr = 2\pi \frac{39}{4}=\frac{39 \pi}{2}, $$

which is what you obtained.