[Math] help me to find the volume of the ice cream cone!

integration

Consider the solid shaped like an ice cream cone that is bounded by the functions enter image description here and enter image description here Set up an integral in polar coordinates to find the volume of this ice cream cone.

I tried to set up the integral but i have not got any write answer!

Best Answer

$z^2=x^2+y^2$ is a cone, and the other one is a sphere.
Volume to be found is intersection of these. find their intersection, its a circle of radius 4.
Now we will setup the integral in cylindrical coordinates, which are also called half polar coordinates.

$$\int\limits_{0}^{2\pi}\int\limits_{0}^{4}\int\limits_{r}^{\sqrt{32-r^2}} r \cdot dz\,dr\,d\theta$$
is the required volume.

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