`int_(0)^(2)x sqrt(2-x)dx`

Explanation

`int_(0)^(2)x sqrt(2-x)dx=int_(0)^(2)(2-x)sqrt(2-(2-x))dx`
`" "[because int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx]`
`=int_(0)^(2)(2-x)sqrtx dx`
`=2int_(0)^(2)x^((1)/(2))dx-int_(0)^(2)x^((3)/(2))dx`
`=(4)/(3).[x^((3)/(2))]_(0)^(2)-(2)/(5)[x^((5)/(2))]_(0)^(2)`
`=(8sqrt2)/(3)-(8sqrt2)/(5)=(16sqrt2)/(15)`