[解 1]
積分領域が四分円なので, 極座標変換します。 即ち
x = r cosθ, y = r sinθ, 0 < r ≦ 1, 0 ≦ θ ≦ π/2.
Jacobian は ∂(x, y)/ ∂(r, θ) = r なので,
与式 = ∫_(0 < r ≦ 1, 0 ≦ θ ≦ π/2) \(r^{2}\) cosθdr dθ
= [\(r^{3}\)/3]_\(0^{1}\)[sinθ]_0^(π/2) = \(\frac{1}{3}\).

[解 2]
与式 = ∫_(y = 0\()^{1}\)∫_(x = 0)^(\(\sqrt{\quad}\)(1 - \(y^{2}\))) x dx dy
= ∫_(y = 0\()^{1}\) [\(x^{2}\)/2]_(x = 0)^(\(\sqrt{\quad}\)(1 - \(y^{2}\))) dy
= (\(\frac{1}{2}\)) ∫_(y = 0\()^{1}\) (1 - \(y^{2}\)) dy
= (\(\frac{1}{2}\)) [y - \(y^{3}\)/3]_\(0^{1}\)
= (\(\frac{1}{2}\))(\(\frac{2}{3}\)) = \(\frac{1}{3}\).