Question

The area bounded by $f(x)=x^2,0\le x\le 1,g\left(x\right)=x+2,1\le x\le 2$ and $x$- axis is

• A. $\frac{3}{2}$
• B. $\frac{4}{3}$
• C. $\frac{8}{3}$
• D. None of these

Answer 1

Answer: (D)

Required area $=$ Area of $OAB+$ Area of $ABC$
Now, Area of $OAB=\underset{0}{\overset{1}{\int }}f(x)dx+\underset{1}{\overset{2}{\int }}g(x)dx$
$=\underset{0}{\overset{1}{\int }}{x}^{2}dx+{\int }_1^2(-x+2)dx$
$=\frac{x^3}{3}{1}_0^1+{\left[\frac{-x^2}{2}+2x\right]}_1^2$
$=\frac{1}{3}+\left[\left(\frac{-4}{2}+4\right)-\left(\frac{-1}{2}\right)+2\right]$
$=\frac{1}{3}+\left[(-2+4)-\left(\frac{3}{2}\right)\right]$
$=\frac{1}{3}+\frac{1}{2}=\frac{5}{6}$ sq unit