Question

The area of the region bounded by the curve $y=x^2+1$ and $y=2x-2$ between $x=-1$ and $x=2$ is:

• A. $9$sq. units
• B. $12$sq. units
• C. $15$sq. units
• D. $14$sq. units

Answer 1

The correct option is A $9$sq. units

We have to find area of the region bounded by curves $y=x^2+1$ and $y=2x-2$ between $x=-1andx=2$
To find points of intersections, if any, for the parabola and the straight line we solve both simultaneously.
$x^2+1=2x-2$
$\Rightarrow x^2-2x+3=0$, which has no real solutions. Hence, no points of intersection for the parabola and the straight line.
From the figure, the graph of $y=x^2+1$ will be always above the graph of $y=2x-2$.
Hence the required area is $\stackrel{2}{\underset{-1}{\int }}\left[\right(x^2+1)-(2x-2\left)\right]dx$

$\stackrel{2}{\underset{-1}{\int }}(x^2-2x+3)dx$

$=\frac{x^3}{3}{|}_{-1}^2-x^2{|}_{-1}^2+3x{|}_{-1}^2$

$=3-\left(3\right)+9$

$=9squnits.$