Below the sphere x 2y z2 = 1, using cylindrical or spherical coordinates, whichever seems more appropriate Recall that the centroid is the center of mass of the solid assuming constant density Solution In spherical coordinates, the regions are given by 0 ˚ ˇ=4;0 ˆ 1 Thus, we compute the volume in spherical coordinates Vol(E) = ZZZ EUse spherical coordinates to evaluate the triple integral e−x2−y2−z2 x2y2z2 dV, ∫∫∫ E where E is the region bounded by the spheres x2y2z2=25 and x2y2z2=81 Change to spherical coordinates ρ2=25 ρ2=81 (5 pts) Set up the Integral e−ρ2 ρ ρ2sinφdρdφdθ ρ=5 ∫9 φ=0 ∫π θ=0 ∫2π (5 pts) (5 pts) Evaluate the IntegralExample 5 Find the z coordinate of the center of mass of the solid consisting of the part of the hemisphere z = p 4 x 2 y2 inside the cylinder x y2 = 2x if the density ˆ = 1 Answer Again we try using cylindrical coordinates, this time from the start Note that x2 y2 = 2x is not centered at the origin x y = 2x z = 4 x y 2 2 2 2 x y z
Cylindrical And Spherical Coordinates Calculus Volume 3
