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We have a conical solid bounded by the surface:

$z=2 \sqrt{x^{2}+y^{2}}$ and $z=2$

where $R=1$ and $H=2$

set up the integral in:

1) Cartesian (in the order $dzdydx$)

2) Cylindrical (in the order $dzdrd\theta$)

3) Spherical (in the order $d\rho d\phi d\theta$)

I think I understand the basics of this question but am a little lost on the bounds and how exactly each one changes from plane to plane. If someone could run through this easier example with me I am hoping to figure out a harder version involving a sphere inside of a cylinder.

THANK YOU IN ADVANCE!!

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  • $\begingroup$ see adrian banner princeton ocw on multivariable calculus (youtube) .he has explained these types beautifully $\endgroup$ Nov 10 '14 at 9:22
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Hints:

(1) Let $D$ the disk with border $2=2\sqrt{x^{2}+y^{2}}$ (why this disk?). the integral is $$\iint_D\int_{z=\text{floor}}^{z=\text{ceiling}}f dz\,dxdy$$ What is floor and ceiling?

(2) Same idea. What is $D$ in cylindrical coordinates?

Edit your message and post your calculations.

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