Finding Volume by use of Triple Integrals

[maiad]

Homework Statement

Find the Volume of the solid eclose by y=x$^{2}$+z$^{2}$ and y=8-x$^{2}$-z$^{2}$

The Attempt at a Solution

Well know they're both elliptic paraboloids except one is flipped on the xz-plane and moved up 8 units. Knowing this, i equated the two equations and got 4=x$^{x}$+z$^{2}$ which is the Domain of this volume.

I found the limits of the intergration... x$^{2}$+z$^{2}$$\leq$y$\leq$8-x$^{2}$-z$^{2}$...-$\sqrt{y-z^{2}}$$\leq$x$\leq$$\sqrt{y-z^{2}}$...-2$\leq$z$\leq$2

But my problem now is what am i actually integrating. would it be x$^{2}$+z$^{2}$?

 

[LCKurtz]

Homework Statement

Find the Volume of the solid eclose by y=x$^{2}$+z$^{2}$ and y=8-x$^{2}$-z$^{2}$

The Attempt at a Solution

Well know they're both elliptic paraboloids except one is flipped on the xz-plane and moved up 8 units. Knowing this, i equated the two equations and got 4=x$^{x}$+z$^{2}$ which is the Domain of this volume.

I found the limits of the intergration... x$^{2}$+z$^{2}$$\leq$y$\leq$8-x$^{2}$-z$^{2}$...-$\sqrt{y-z^{2}}$$\leq$x$\leq$$\sqrt{y-z^{2}}$...-2$\leq$z$\leq$2

But my problem now is what am i actually integrating. would it be x$^{2}$+z$^{2}$?

No. The integrand would be $1$.$$
Vol = \iiint_R 1\, dV$$And I would suggest you using polar coordinates in the $xz$ plane after you do the $dy$ integral.