∬D11−x2−y2 dxdyD:x2+y2≤1\iint_D \frac{1}{\sqrt{1-x^2-y^2}}\,dxdy \qquad D: x^2+y^2 \leq 1∬D1−x2−y21dxdyD:x2+y2≤1 x=rcosθy=rsinθx = r\cos\theta \quad y = r\sin\thetax=rcosθy=rsinθ とおくと 0≤r≤10≤θ≤2π0 \leq r \leq 1 \quad 0 \leq \theta \leq 2\pi0≤r≤10≤θ≤2π ∫02π∫0111−r2 r dr dθ=∫02π[−1−r2]01dθ=∫02πdθ=2π\int_0^{2\pi}\int_0^1 \frac{1}{\sqrt{1-r^2}}\,r\,dr\,d\theta = \int_0^{2\pi}\left[-\sqrt{1-r^2}\right]_0^1 d\theta = \int_0^{2\pi} d\theta = 2\pi∫02π∫011−r21rdrdθ=∫02π[−1−r2]01dθ=∫02πdθ=2π