Integrating over Region A: Calculating the Volume of a Cone

[WannaBe22]

Homework Statement

Let A be the region that in space bounded by the balls:
$$x^2 +y^2 + z^2 =1$$ , $$x^2 +y^2 +z^2 =4$$ , above the plane $$z=0$$ and inside the cone $$z^2 = x^2 +y^2$$.

A. Write the integral $$\int \int \int_{A} f(x,y,z) dxdydz$$ in the form:
$$\int \int_{E} (\int_{g^1(x,y)}^{g^2(x,y)} f(x,y,z) dz) dxdy$$ when :
$$A=( (x,y,z) | (x,y) \in E, g^1(x,y) \le z \le g^2(x,y) )$$ ...

B. Find the volume of A (not necessarily using part A).

Hope you'll be able to help me in this... I think the main problem is that I can't figure out how A looks like... There is also a hint that one of the functions g1 or g2 should be defined at a split region... I can't figure out how this cone looks like and how I can describe A as equations ...


Thanks in advance!

Homework Equations

The Attempt at a Solution

 

[lanedance]

so you have a sphere of radius 1 & and a sphere of radius 2...

to visualise the cone z^2 = y^2 + x^2, consider the coordinate planes y = 0 and x = 0, these will be vertical slices through the cone - what is the equation of the curve in each plane.

no consider a slice for a given z, you have y^2 + x^2 = z^2, which is an equation for a circle in x & y - what is the radius?

the 2nd part will be much easier to do in spherical coordinates...