Volume of the space region inside sphere outside coni

[melihaltintas]

Homework Statement

Hi everybody i have a problem please help me (sorry for my bad english)

Homework Equations

volume of the space region inside sphere outside coni
z^2=x^2+y^2 coni
x^2+y^2+z^2 =1 sphere

The Attempt at a Solution

I am new in this forum , I search question like this and i found but i didn't solve this question with method that is performed other questions

 

[DryRun]

Hi melihaltintas

First, always plot the graph so you can understand how to find the limits.

I have attached the graph of the xz-trace coordinates (in the plane y = 0). The region of volume that you need to find is shaded in blue.

Start by finding the points of intersection of the sphere and cones (in the plane y = 0).

Your equations for the cones and sphere in the plane y = 0, become:
$$z^2=x^2<br /> \\x^2+z^2 =1$$
Solve these two equations to find the x and z. From there, you'll be able to find the two required angles for $\phi$ which define the limits.

Finally, use the triple integral formula in terms of spherical coordinates:
$$\int \int \int \rho^2 \sin \phi d\rho d\phi d\theta$$

 

[melihaltintas]

thanks a lot :)