The value of `int_(-2)^2|1-x^2|dx` is_______

Options

A. `(1)/(3)`
B. `(14)/(3)`
C. `(7)/(3)`
D. (28)/(3)`

Answer

D

Explanation

We have,
`|1-x^(2)|=|x^(2)-1|={:{((x^(2)-1)"if"-1 lt x lt 1),(" "x^(2)-1","" otherwise"):}`
`:. I=overset(3)underset(-2)int |1-x^(2)|dx`
`rArr I=overset(1)underset(2)int (x^(2)-1)dx+overset(1)underset(-1)int -(x^(2)-1)dx+overset(3)underset(1)int (x^(2)-1)dx`
`rArr I=[(x^(3))/(3)-x]_(-2)^(-1)-[(x^(3))/(3)-x]_(-1)^(1)+[(x^(3))/(3)-x]_(1)^(3)`
`rArr I=[(-(1)/(3)+1)-(-(8)/(3)+2)]-[((1)/(3)-1)-((1)/(3)-1)]+[((27)/(3)-3)-((1)/(3)-1)]`
`rArr I=(4)/(3)+(4)/(3)+(20)/(3)=(28)/(3)`