Find the volume of the solid enclosed by the paraboloids and .
I have no idea how to use integrals to solve this!
You have to draw the region first. See below. We're going from the brown which is the surface $\displaystyle 25(x^2+y^2)$ to the blue which is $\displaystyle 8-25(x^2+y^2)$. Since it's radially symmetrical, we can calculate just the volume over the first quadrant and then multiply by 4:
$\displaystyle V=4\int_0^{\sqrt{8/50}}\int_0^{\sqrt{8/50-x^2}}\int_{25(x^2+y^2)}^{8-25(x^2+y^2)} dzdydx$
May want to convert to spherical coordinates.
In cylindrical coordinates those are $\displaystyle z= 25 r^2$ and $\displaystyle z= 8- 25 r^2$. They intersect where $\displaystyle 25r^2= 8- 25r^2$ or [tex]50r^2= 8[tex] which has solution r= 2/5. The volume is given by
$\displaystyle \int_{\theta= 0}^{2\pi}\int_{r= 0}^{2/5} (8- 50 r^2)(r dr d\theta)$