Use spherical coordinates to find the volume of the region outside the cone and inside the sphere. We set the triple integral up using spherical coordinates.
Use spherical coordinates to find the volume of the region outside the cone and inside the sphere. We set the triple integral up using spherical coordinates. Mar 28, 2025 · We wish to find the volume of the region in spherical coordinates that lies outside the cone φ = 4π and inside the sphere defined by ρ = 4cosφ. Solution Attempt: I can visualize the surfaces and see that the volume is two spherical caps at the edges of the cone but am not sure how to set up the integral. Because the region is described as "outside the cone φ = π/4," the polar angle φ ranges from φ = 4π to φ = 2π, In this video we compute the volume contained inside a sphere, outside a cone, and above the xy-plane using two approaches. Feb 20, 2025 · This video shows how to find the volume inside a sphere and outside a cone. May 30, 2019 · Using a volume integral and spherical coordinates, we derive the formula of the volume of the inside of a sphere, the volume of a ball. In spherical coordinates, the volume element is given by: Nov 26, 2023 · The volume of the region outside the cone and inside the sphere is calculated using spherical coordinates, resulting in the formula 31πρ3. May 30, 2015 · Use spherical coordinates to find the volume of the region lying above z = 3x2 + 3y2− −−−−−−−√ z = 3 x 2 + 3 y 2 and within the x2 +y2 +z2 = 2az x 2 + y 2 + z 2 = 2 a z, a> 0 a> 0. So far I know that the first graph is a cone and the second one is some kind of sphere. To find the volume of the region outside the cone ϕ = 4π and inside the sphere ρ = 4cosϕ, we utilize spherical coordinates. Inside is an IMAX screen that changes the sphere into a planetarium with a sky full of \ (9000\) twinkling stars. The maximum value of ρ occurs at φ=0 where cosφ=1. Mar 27, 2025 · For example, consider a sphere with radius defined by the equation given above and rotating around the z-axis while extending beyond the cone's surface. Set up the triple integral using spherical coordinates that should be used to find the volume as efficiently as possible. Using triple integrals in spherical coordinates, we can find the volumes of different geometric shapes like these. Just imagine shooting a hole through a sphere, and then finding the volume of what remains. I have completed the square so that the new equation is: Jun 28, 2015 · The region outside the cylinder and inside the sphere doesn't include the end caps. Therefore, the sphere has a radius that diminishes as we move . In spherical coordinates, the volume element is given by dV = ρ2 sinφdρdφdθ. This visualization allows students to comprehend how three-dimensional shapes interact within certain bounds defined in spherical coordinates. Use spherical coordinates to find the volume of the region outside the cone phi = pi/4 and inside the sphere rho = 8 cos phi. First we use cylindrical coordinates integrating r first. Use spherical coordinates to find the volume of the region outside the cone phi = pi/4 and inside the sphere rho = 11 cos phi. Therefore, the chosen multiple-choice option is A. Mastering the integration limits based on the shapes involved is key to solving such problems. " Mar 27, 2025 · To find the volume of the region outside the cone φ=4π and inside the sphere ρ=4cosφ using spherical coordinates, we will follow these steps: Firstly, let’s analyze the region we are investigating: Sphere: The equation ρ=4cosφ describes a sphere in spherical coordinates. nwhktktaoonynzejcvfvftofpnjzqsvbmpmgdklxptrwpljds