Double Integration to Find Volume

[doppelganger007]

Homework Statement

Find the volume of the region inside the surface z=x^2+y^2 and between z=0 and z=10

Homework Equations

x^2+y^2=10

The Attempt at a Solution

I know that I have to use some sort of double integration to find this volume, but I'm not really sure where to begin with the problem

 

[Kummer]

\iiint_V dV the upper curve is z=10 the lower curve is z=0 the area of integration in \mathbb{R}^2 is a circle of radius 10. Now use cyclindrical change of variable.

 

[doppelganger007]

hmm...I may try that if I can't find an alternative, but is there any way to do this problem with a double integration and not a triple integration? because the section I'm working with is strictly double integration

 

[HallsofIvy]

That's a paraboloid. Since you have "flat" bottom and top I recommend you imagine the solid consisting of thin horizontal pieces. It should be obvious that the piece at height z is a disk satisfying x^2+ y^2= z. What is the area of that disk? If you think of the disk as having thickness "dz", what is its volume? Now "add" the volumes of all those disks.