The slope of the tangent to a curve `y=f(x)` at `(x,f(x))` is `2x+1.` If the curve passes through the point `(1,2)` then the area of the region bounded by the curve, the x-axis and the line `x=1` is (A) `5/6` (B) `6/5` (C) `1/6` (D) `1`

Options

A. `5//6`
B. `6//5`
C. `6`
D. `1`

Answer

A

Explanation

Slope of tangent
`dyh//dx=2x+1`
`impliesy=x^(2)+x+c`
When `x=1,y=2`
`:.2=1+1+CimpliesC=0`
`:.y=x^(2)+x`
`:.` Required area `=int_(0)^(1)(x^(2)+x)dx`

`=[(x^(3))/3+(x^(2))/2]_(0)^(1)=5//6`