Someone computes the volume of a solid sphere of radius in spherical coordinates as…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Multiple integrals and change of variables

Problem

Someone computes the volume of a solid sphere of radius RR in spherical coordinates as follows: "V=∫02π∫0π∫0Rr2 dr dθ dϕ=2π×π×R33=2π2R33.V = \displaystyle\int_0^{2\pi}\int_0^{\pi}\int_0^{R} r^2\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi = 2\pi\times\pi\times\dfrac{R^3}{3} = \dfrac{2\pi^2R^3}{3}." They then use this volume to find the mass of a solid sphere of radius R=0.25 mR = 0.25\,\mathrm{m} with uniform density ρm=2700 kg m−3\rho_{\mathrm{m}} = 2700\,\mathrm{kg\,m^{-3}} (aluminium), as M=ρmVM = \rho_{\mathrm{m}}V. ((a)) Find the error in the working and explain what it changes. ((b)) Give the correct volume and the correct mass, to two significant figures.

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