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!