Question

The area (in sq units) of the region bounded by $x=-1,x=2,y=x^2+1$ and $y=2x-2$is

• A. $10$
• B. $7$
• C. $8$
• D. $9$

Answer 1

The correct option is D

$9$

Explanation for the correct option:

The area of bounded region:

The curves which are given in the equation are :

$$\begin{gathered} x=-1, \\ x=2, \\ y=x^2+1 \end{gathered}$$ .

$y=2x-2$

Now the area of the region bounded by these are,

$$\begin{gathered} \Rightarrow A={\int }_{-1}^2(x^2+1)-(2x-2)dx \\ \Rightarrow A={\int }_{-1}^2(x^2+1-2x+2)dx \\ \Rightarrow A={\int }_{-1}^2(x^2-2x+3)dx \\ \Rightarrow A={[\frac{x^3}{3}-x^2+3x]}_{-1}^2 \\ \Rightarrow A=[\frac{8}{3}-4+6-(-\frac{1}{3}-1-3] \\ \Rightarrow A=[\frac{8}{3}-4+6+\frac{1}{3}+1+3] \\ \Rightarrow A=[\frac{8}{3}+\frac{1}{3}+6] \\ \Rightarrow A=\frac{8+1+18}{3} \\ \Rightarrow A=\frac{27}{3} \\ \Rightarrow A=9sq.units \end{gathered}$$

Therefore, the area bounded by the given curves is equal to $9$ sq.units

Hence, option (D) is the correct answer.