Question

The area of the region bounded by the curve y=2x–$x ^ { 2 }$ and line y =x is

• A. $\frac { 1 } { 2 }$
• B. $\frac { 1 } { 3 }$
• C. $\frac { 1 } { 4 }$
• D. $\frac { 1 } { 6 }$

Answer 1

Answer: (D)

$\frac { 1 } { 6 }$

Answer 2

The given curve is $y = 2 x - x ^ { 2 }$

$\Rightarrow y = - \left( x ^ { 2 } - 2 x + 1 \right) + 1$

$\Rightarrow y - 1 = - ( x - 1 ) ^ { 2 }$ it represents a downward parabola with vertex (1,1)

Its points of intersection with the line y = x are (0,0) and (1,1).

Required area = shaded region

$= \int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x - \int _ { 0 } ^ { 1 } x d x$ $= \int _ { 0 } ^ { 1 } \left( x - x ^ { 2 } \right) d x = \left( \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 3 } } { 3 } \right) _ { 0 } ^ { 1 }$

$= \frac { 1 } { 2 } - \frac { 1 } { 3 } = \frac { 1 } { 6 }$ .