First we will evaluate: ( substitution: u = x² - y², du = - 2 y dy )
[tex] \int\limits^x_0 {y \sqrt{ x^{2} - y^{2} } } \, dy= \\ \frac{-1}{2} \int\limits^x_0 { u^{1/2} } \, du =[/tex]
=[tex] \frac{-1}{3} \sqrt{ (x^{2} - y^{2} ) ^{3} } [/tex] ( than plug in x and 0 )
=[tex]- \frac{1}{3} ( \sqrt{( x^{2} - x^{2}) ^{3} } - \sqrt{ (x^{2} -0 ^{2} ) ^{3} } [/tex] =
= 1/3 x³ ( then another integration )
[tex] 1/3\int\limits^1_0 { x^{3} } \, dx = 1/3 ( x^{4}/4)}= 1/3 ( 1 ^{4}/4 - 0^{4} /4 ) [/tex]
= 1/3 * 1/4 = 1/12
