Question

 Evaluate: `int_-6^3 |x+3|dx`

Answer 1

`I = int_-6^3|x+3|dx`
`f(x)=|x+3|=X+3:  X+3>0rArr X > -3`

                = -x -3 ; x+3<0 ⇒ x < -3

`I = int_-6^3 |x+3| dx + int_-3^3(x+3)dx`

`I = -(x^2/2 +3x)_-6^-3 + (x^2/2+3x)_-3^3`

`I = - [(9/2 - 9)-(36/2 -18)]+[(9/2+9)-(9/2 - 9)]`

`I = -[-9/2]+18`

`I = 9/2 +18 = (9+36)/2 = 45/2`
Alternate method :

`int_-6^3 |x+3| dx `
= `A(DeltaABC)+A(DeltaCDE)`

`= 1/2xx3xx3+1/2xx6xx6`

`=9/2+18`

`=45/2`