The discussion focuses on calculating the volume bounded by the sphere defined by the equation x² + y² + z² = 4 and the exterior of the cylinder described by x² + y² = 2x. Participants suggest using cylindrical coordinates for integration, specifically the limits ∫(θ=0,2π)dθ∫(r=1,2)dr∫(z=0,sqrt(4-r²)). They emphasize the importance of accurately determining the limits for z, r, and θ, particularly where the sphere intersects the cylinder. The final volume removed by the cylinder is approximately 9.644, leaving a volume of 23.85 in the sphere.
PREREQUISITESMathematicians, physics students, and engineers involved in volume calculations and geometric analysis, particularly those working with integrals in three-dimensional space.
haruspex said:
I came to the same conclusionDick said:
. In the end, I did it with a single integration. In the z plane, the area to be removed is the intersection of two circles. That turns into the sum of two sectors minus the sum of two triangles, so I could write those straight down. Final result is the removal of 16π/3-64/9, leaving 16π/3+64/9. Neatest way to express it is that it leaves 16/9 in the hemisphere it lies in.