What is the area of the region bounded by x-asix, the curve y=f(x) and the two ordinates `x=(1)/(2)andx=1`?

Options

A. `(5)/(12)` square unit
B. `(5)/(6)` square unit
C. `(7)/(6)` square units
D. 2 square units

Answer

A

Explanation

`f(x)={{:(x^(2)+x-1,xge1,AA,x""inR),(x^(2)-x+1,xlt1,,):}`
Hence area required for given region is
`A_(1)=underset(1//2)overset(1)intf(x)dx`
`=underset(1//2)overset(1)int(x^(2)-x+1)dx`
`=[(1)/(3)x^(3)-(x^(2))/(2)+x]_(1//2)^(1)`
`=((1)/(3)xx1-(1)/(2)+1)-((1)/(3)xx(1)/(8)-(1)/2xx(1)/(4)+(1)/(2))`
`A_(1)=(5)/(12)` square unit.