Someone evaluates

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

Conditions of Validity, Ch Multiple integrals: Fubini, the change-of-variables theorem and integrals that do not exist

Problem

Someone evaluates I=∬Sx dAI=\displaystyle\iint_{S} x\,\mathrm{d}A, where S=[0,1]×[0,1]S=[0,1]\times[0,1] is the unit square, by changing to plane polar coordinates x=rcos⁡θx=r\cos\theta, y=rsin⁡θy=r\sin\theta. Their working reads: "dA=r dr dθ\mathrm{d}A = r\,\mathrm{d}r\,\mathrm{d}\theta, and the square SS corresponds to 0≤r≤10\le r\le1, 0≤θ≤π/20\le\theta\le\pi/2. So I=∫0π/2 ⁣ ⁣∫01(rcos⁡θ) r dr dθ=∫0π/2cos⁡θ dθ⋅∫01r2 dr=1⋅13=13.I = \int_0^{\pi/2}\!\!\int_0^1 (r\cos\theta)\,r\,\mathrm{d}r\,\mathrm{d}\theta = \int_0^{\pi/2}\cos\theta\,\mathrm{d}\theta \cdot \int_0^1 r^2\,\mathrm{d}r = 1\cdot\tfrac13=\tfrac13." ((a)) Find the error in this working. ((b)) State the correct value of II.

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