# Thread: Need Help in Vector Calculus Multiple Integral

1. ## Need Help in Vector Calculus Multiple Integral

Use cylindrical coordinates to find the volume of the solid between two sheets of the hyperboloid z2 =64+x2+y2 bounded by the cylinder x2+y2=225. (HINT: Use symmetry and elementary geometry to simplify the calculation.)

2. ## Re: Need Help in Vector Calculus Multiple Integral

What help do you need? Have you converted the hyperboloid to cylindrical coordinates?

3. ## Re: Need Help in Vector Calculus Multiple Integral

$\displaystyle 225= 15^2$ so $\displaystyle x^2+ y^2= 225$, $\displaystyle r= 15$ in cylindrical coordinates, is the cylinder with axis the z-axis and radius 15. The hyperboloid is $\displaystyle z^2= 64+ x^2+ y^2= 64+ r^2$ or $\displaystyle z= \pm\sqrt{64+ r^2}$ in cylindrical coordinates. The volume of the region between the two branches of the hyperbola, inside the cylinder, is given by $\displaystyle \int_0^{2\pi}\int_0^{15}\int_{-\sqrt{64+ r^2}}^{\sqrt{64+ r^2}} r dzdrd\theta}= 2\pi\int_0^{15}\int_{-\sqrt{64+ r^2}}^{\sqrt{64+ r^2}} r dzdr= 4\pi\int_0^{15} r\sqrt{64+ r^2}dr$.

4. ## Re: Need Help in Vector Calculus Multiple Integral

Thanks a lot!