x=rcosθx = r\cos\thetax=rcosθ y=rsinθy = r\sin\thetay=rsinθ とおくと 0≤r≤a0 \leq r \leq a0≤r≤a 0≤θ≤π20 \leq \theta \leq \dfrac{\pi}{2}0≤θ≤2π ∫0π2∫0artan−1rsinθrcosθ drdθ=∫0π2∫0artan−1(tanθ) drdθ=∫0π2∫0arθ drdθ\int_0^{\frac{\pi}{2}} \int_0^a r \tan^{-1} \frac{r\sin\theta}{r\cos\theta} \, dr d\theta = \int_0^{\frac{\pi}{2}} \int_0^a r \tan^{-1}(\tan\theta) \, dr d\theta = \int_0^{\frac{\pi}{2}} \int_0^a r\theta \, dr d\theta∫02π∫0artan−1rcosθrsinθdrdθ=∫02π∫0artan−1(tanθ)drdθ=∫02π∫0arθdrdθ =∫0π2[12r2θ]0adθ=∫0π212a2θ dθ=[14a2θ2]0π2=π216a2= \int_0^{\frac{\pi}{2}} \left[ \frac{1}{2} r^2 \theta \right]_0^a d\theta = \int_0^{\frac{\pi}{2}} \frac{1}{2} a^2 \theta \, d\theta = \left[ \frac{1}{4} a^2 \theta^2 \right]_0^{\frac{\pi}{2}} = \frac{\pi^2}{16} a^2=∫02π[21r2θ]0adθ=∫02π21a2θdθ=[41a2θ2]02π=16π2a2