Finding the volume of the solid using change of variables technique

[Dhamnekar Win,odd]

Find the volume of the solid bounded by [math]z = x^2  + y^2 [/math] and [math] z^2 = 4(x^2 + y^2) [/math]

My attempt to answer this question:

[math] \displaystyle\int_0^{2\pi}\displaystyle\int_0^2\displaystyle\int_0^4 1 \rho^2 \sin{(\phi)} d\rho d\phi d\theta = -\frac{128\pi \cos{(2)}- 128\pi}{3}= 189.822143919[/math] Is this answer correct?

 

[Dhamnekar Win,odd]

 I think my answer in the original post is incorrect. So, I want to correct it. My corrected answer is [math] \displaystyle\int_0^{2\pi}\displaystyle\int_0^2\displaystyle\int_0^4 r dz dr d\theta =16\pi [/math] Is this answer correct? 

Edited June 6, 2022 by Dhamnekar Win,odd

[Dhamnekar Win,odd]

Answers to all of the aforesaid questions are affirmative.