Question

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`
A. `(3)/(2)`
B. `(4)/(3)`
C. `(5)/(6)`
D. `(1)/(12)`

Answer 1

Correct Answer - C
Given, `(dy)/(dx)=2x+1`
On integrating both sides
` int dy=int(2x+1)dx `
` rArr y=x^(2)+x+C ` which passes passes through(1,2)
` therefore 2=1+1+C `
` rArr C=0`
` therefore y=x^(2)+x `

Thus, the required area bounded by X-axis, the curve and x=1
`=int_(0)^(1)(x^(2)+x)dx=[x^(3)/(3)+(x^(2))/(2)]_(0)^(1)`
`=(1)/(3)+(1)/(2)=(5)/(6)` sq unit