find the volume the the solid in ~ the ball $x^2+y^2+z^2=9$, outside the cone $z=\sqrtx^2+y^2$, and above the $xy$-plane.

You are watching: Find the volume of the solid that is enclosed by the cone and the sphere

Using Cylindrical coordinates, $r^2+z^2=9$ and also $z=r$. Intersection is $x^2+y^2=\frac92$

\beginalignV=\int_0^2\pi\int_\frac3\sqrt 2^3\int_0^\sqrt9-r^2r\:dz\:dr\:d\theta+\int_0^2\pi\int_0^\frac3\sqrt2\int_0^r r\:dz\:dr\:d\theta\endalignAgain utilizing Spherical coordinates,I can"t figure out $\phi$ and also $\rho$. I simply need those border anyone have the right to skip evaluation. And is my an initial approach correct$?$ Any help will it is in appreciated.Thanks in advances.

Using the complying with substitutions because that spherical coordinates:$$z = \rho \cos(\phi)$$$$x = \rho \sin(\phi)\cos(\theta)$$$$y = \rho \sin(\phi)\sin(\theta)$$\beginalign\rho \cos(\phi)&=\sqrt\rho^2 \sin^2(\phi)\cos^2(\theta)+\rho^2 \sin^2(\phi)\sin^2(\theta)\\&=\sqrt\rho^2\sin^2(\phi)\\\phi&=\frac\pi4\endalignAbove the $xy$-plane hence $\phi\leq \frac\pi2\implies \frac\pi4\leq\phi\leq\frac\pi2$$\int_0^2\pi\int_\frac\pi4^\frac\pi2\int_0^3\rho^2 \sin(\phi) d\rho d\phi d\theta$$You deserve to think the volume is totality upper hemisphere except the cone that"s why$0\leq \theta\leq 2\pi$and also$0\leq \rho\leq 3$Thanks because that contributing response to naipublishers.comematics ridge Exchange! Please be sure to answer the question. Provide details and also share her research! But avoid Asking for help, clarification, or responding to various other answers.Making statements based on opinion; back them up with referrals or an individual experience. Use naipublishers.comJax to format equations. Naipublishers.comJax reference. See more: Mut 500 Madden 17 Codes Xbox One (X1), Mut 500 Madden Points Pack To find out more, watch our tips on writing good answers. article Your price Discard By click “Post her Answer”, girlfriend agree to our terms of service, privacy policy and cookie plan ## Not the answer you're looking for? Browse other questions tagged multivariable-calculus or asking your own question. discover the volume because that the region that remains in the spherical hard$\rho \leq 4$after the hard cone$\phi \leq \pi/6$has actually been removed? discover volume that the solid which is the intersection of the solid sphere$x^2+y^2+z^2 \leq 9$and the solid cylinder$x^2+y^2\leq 1\$

site style / logo © 2021 ridge Exchange Inc; user contributions licensed under cc by-sa. Rev2021.10.1.40358

naipublishers.comematics ridge Exchange works best with JavaScript permitted